{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2025-12-14T01:33:56.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2025-12-14T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":45979,"title":"Compute the perimeter of an ellipse","description":"While the area of an ellipse is straightforward to compute, the perimeter (or circumference) is more complicated. The perimeter can be expressed in terms of elliptic integrals, and several approximate formulas are available as well. \r\n\r\nCompute the perimeter of an ellipse given the lengths of the semi-major and semi-minor axes--in any order.","description_html":"\u003cp\u003eWhile the area of an ellipse is straightforward to compute, the perimeter (or circumference) is more complicated. The perimeter can be expressed in terms of elliptic integrals, and several approximate formulas are available as well.\u003c/p\u003e\u003cp\u003eCompute the perimeter of an ellipse given the lengths of the semi-major and semi-minor axes--in any order.\u003c/p\u003e","function_template":"function P = ellipsePerim(a,b)\r\n y = f(a,b);\r\nend","test_suite":"%%\r\na = 3; \r\nb = 4;\r\nP_correct = 22.103492160709504;\r\nassert(abs(ellipsePerim(a,b)-P_correct)/P_correct\u003c1e-8)\r\n\r\n%%\r\na = 4; \r\nb = 3;\r\nP_correct = 22.103492160709504;\r\nassert(abs(ellipsePerim(a,b)-P_correct)/P_correct\u003c1e-8)\r\n\r\n%%\r\na = 1;\r\nb = 8;\r\nP_correct = 32.744956600195508;\r\nassert(abs(ellipsePerim(a,b)-P_correct)/P_correct\u003c1e-8)\r\n\r\n%% \r\na = 1;\r\nb = 0.974062207869516;\r\nP_correct = 6.201967;\r\nassert(abs(ellipsePerim(a,b)-P_correct)/P_correct\u003c1e-8)\r\n\r\n%%\r\na = 4*rand(1);\r\nb = a;\r\nP_correct = 2*pi*a;\r\nassert(abs(ellipsePerim(a,b)-P_correct)/P_correct\u003c1e-8)","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":26,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-06-22T14:29:06.000Z","updated_at":"2026-01-02T12:52:12.000Z","published_at":"2020-06-22T16:17:57.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWhile the area of an ellipse is straightforward to compute, the perimeter (or circumference) is more complicated. The perimeter can be expressed in terms of elliptic integrals, and several approximate formulas are available as well.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCompute the perimeter of an ellipse given the lengths of the semi-major and semi-minor axes--in any order.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":59147,"title":"Determine aquifer properties: steady pump test in a leaky confined aquifer","description":"Problem statement\r\nWrite a function to determine the hydraulic conductivity of a leaky confined aquifer and the hydraulic conductivity of the leaky confining layer given the pumping rate and radius of the well, the drawdowns measured at distances from the well, and the thicknesses and of the leaky confined aquifer and the leaking confining layer, respectively. \r\nBackground\r\nCody Problem 59002 dealt with one-dimensional flow in a leaky confined aquifer. This problem involves flow to a well in a leaky confined aquifer. As in other pumping tests, the idea is to determine the properties of the aquifer by disturbing it, observing the response, and comparing to an analytical solution. Here the two unknown hydraulic conductivities are determined from two observations of drawdown.\r\nAn analytical solution for the drawdown can be determined by solving the equation\r\n\r\nwith the boundary conditions that the drawdown far from the well is zero (i.e., ) and the flow at the well is . Using Darcy’s law and the convention that flow to the well is positive, one finds\r\n\r\nThe governing equation is related to the one in Cody Problem 51783, and it is the polar coordinates version of the equation in Cody Problem 59002. \r\n","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 819.1px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 409.55px; transform-origin: 407px 409.55px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 64px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 32px; text-align: left; transform-origin: 384px 32px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 171.408px 8px; transform-origin: 171.408px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to determine the hydraulic conductivity \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eK\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 179.325px 8px; transform-origin: 179.325px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e of a leaky confined aquifer and the hydraulic conductivity \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"19\" height=\"18\" alt=\"K'\" style=\"width: 19px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 9.71667px 8px; transform-origin: 9.71667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e of the leaky confining layer given the pumping rate \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"19\" height=\"20\" alt=\"Q0\" style=\"width: 19px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 36.5667px 8px; transform-origin: 36.5667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and radius \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15.5\" height=\"20\" alt=\"rw\" style=\"width: 15.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 86.35px 8px; transform-origin: 86.35px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e of the well, the drawdowns \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003es\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 74.2917px 8px; transform-origin: 74.2917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e measured at distances \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003er\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e from the well, and the thicknesses \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eb\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"18\" alt=\"b'\" style=\"width: 15px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 229.767px 8px; transform-origin: 229.767px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e of the leaky confined aquifer and the leaking confining layer, respectively. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.8333px 8px; transform-origin: 40.8333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eBackground\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/59002\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003eCody Problem 59002\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 310.817px 8px; transform-origin: 310.817px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e dealt with one-dimensional flow in a leaky confined aquifer. This problem involves flow to a well in a leaky confined aquifer. As in \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/49743\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003eother pumping tests\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 215.092px 8px; transform-origin: 215.092px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, the idea is to determine the properties of the aquifer by disturbing it, observing the response, and comparing to an analytical solution. Here the two unknown hydraulic conductivities are determined from two observations of drawdown.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 254.808px 8px; transform-origin: 254.808px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAn analytical solution for the drawdown can be determined by solving the equation\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 36.6px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.3px; text-align: left; transform-origin: 384px 18.3px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"136.5\" height=\"36.5\" alt=\"d2s/dr2 + (1/r)ds/dr - K's/Kbb' = 0\" style=\"width: 136.5px; height: 36.5px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 43px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.5px; text-align: left; transform-origin: 384px 21.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 239.6px 8px; transform-origin: 239.6px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewith the boundary conditions that the drawdown far from the well is zero (i.e., \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"60.5\" height=\"18.5\" alt=\"s(inf) = 0\" style=\"width: 60.5px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 84.4px 8px; transform-origin: 84.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e) and the flow at the well is \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"19\" height=\"20\" alt=\"Q0\" style=\"width: 19px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Using Darcy’s law and the convention that flow to the well is positive, one finds\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 34.8px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 17.4px; text-align: left; transform-origin: 384px 17.4px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"141.5\" height=\"35\" alt=\"Q0 = -2pi rw b K (ds/dr)|_{r=rw}\" style=\"width: 141.5px; height: 35px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 146.267px 8px; transform-origin: 146.267px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe governing equation is related to the one in \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/51783\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003eCody Problem 51783\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 169.983px 8px; transform-origin: 169.983px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and it is the polar coordinates version of the equation in \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/59002\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003eCody Problem 59002\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 370.7px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 185.35px; text-align: left; transform-origin: 384px 185.35px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" width=\"504\" height=\"365\" style=\"vertical-align: baseline;width: 504px;height: 365px\" 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\" alt=\"Steady pump test in a leaky confined aquifer\" data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [K,Kp] = steadyPumpTestLeakyConfined(r,s,Q0,rw,b,bp)\r\n% K = hydraulic conductivity of aquifer; Kp = hydraulic conductivity of confining layer\r\n% Other variables are defined in the test suite\r\n \r\n K = Q0*log(r(1)/r(2))/(2*pi*b*(s(2)-s(1));\r\n Kp = K*bp/b;\r\n \r\nend","test_suite":"%%\r\nr = [30 75]; % Distances from the well (m)\r\ns = [0.4 0.25]; % Observed drawdown (m)\r\nQ0 = 1000; % Pumping rate (m3/d)\r\nrw = 0.6; % Radius of the well (m)\r\nb = 10; % Thickness of the confined aquifer (m)\r\nbp = 1.5; % Thickness of the leaky confining layer (m)\r\n[K,Kp] = steadyPumpTestLeakyConfined(r,s,Q0,rw,b,bp);\r\nK_correct = 93.54;\r\nKp_correct = 1.82e-2;\r\nassert(abs(K-K_correct)/K_correct\u003c1e-4)\r\nassert(abs(Kp-Kp_correct)/Kp_correct\u003c1e-3)\r\n\r\n%%\r\nr = [30 75]; % Distances from the well (m)\r\ns = [0.4 0.25]; % Observed drawdown (m)\r\nQ0 = 2000; % Pumping rate (m3/d)\r\nrw = 0.6; % Radius of the well (m)\r\nb = 10; % Thickness of the confined aquifer (m)\r\nbp = 1.5; % Thickness of the leaky confining layer (m)\r\n[K,Kp] = steadyPumpTestLeakyConfined(r,s,Q0,rw,b,bp);\r\nK_correct = 187.07;\r\nKp_correct = 3.64e-2;\r\nassert(abs(K-K_correct)/K_correct\u003c1e-4)\r\nassert(abs(Kp-Kp_correct)/Kp_correct\u003c1e-3)\r\n\r\n%%\r\nr = [30 75]; % Distances from the well (m)\r\ns = [0.6 0.4]; % Observed drawdown (m)\r\nQ0 = 1000; % Pumping rate (m3/d)\r\nrw = 0.6; % Radius of the well (m)\r\nb = 10; % Thickness of the confined aquifer (m)\r\nbp = 1.5; % Thickness of the leaky confining layer (m)\r\n[K,Kp] = steadyPumpTestLeakyConfined(r,s,Q0,rw,b,bp);\r\nK_correct = 71.32;\r\nKp_correct = 7.00e-3;\r\nassert(abs(K-K_correct)/K_correct\u003c1e-4)\r\nassert(abs(Kp-Kp_correct)/Kp_correct\u003c1e-3)\r\n\r\n%%\r\nr = [30 75]; % Distances from the well (m)\r\ns = [0.6 0.4]; % Observed drawdown (m)\r\nQ0 = 1000; % Pumping rate (m3/d)\r\nrw = 0.3; % Radius of the well (m)\r\nb = 10; % Thickness of the confined aquifer (m)\r\nbp = 3; % Thickness of the leaky confining layer (m)\r\n[K,Kp] = steadyPumpTestLeakyConfined(r,s,Q0,rw,b,bp);\r\nK_correct = 71.32;\r\nKp_correct = 1.40e-2;\r\nassert(abs(K-K_correct)/K_correct\u003c1e-4)\r\nassert(abs(Kp-Kp_correct)/Kp_correct\u003c1e-3)\r\n\r\n%%\r\nr = [100 240]; % Distances from the well (m)\r\ns = [4.0 2.8]; % Observed drawdown (m)\r\nQ0 = 3500; % Pumping rate (m3/d)\r\nrw = 0.3; % Radius of the well (m)\r\nb = 35; % Thickness of the confined aquifer (m)\r\nbp = 2.3; % Thickness of the leaky confining layer (m)\r\n[K,Kp] = steadyPumpTestLeakyConfined(r,s,Q0,rw,b,bp);\r\nK_correct = 11.43;\r\nKp_correct = 3.74e-4;\r\nassert(abs(K-K_correct)/K_correct\u003c1e-4)\r\nassert(abs(Kp-Kp_correct)/Kp_correct\u003c1e-3)\r\n\r\n%%\r\nr = [100 240]; % Distances from the well (m)\r\ns = [10 8]; % Observed drawdown (m)\r\nQ0 = 3500; % Pumping rate (m3/d)\r\nrw = 0.3; % Radius of the well (m)\r\nb = 35; % Thickness of the confined aquifer (m)\r\nbp = 2.3; % Thickness of the leaky confining layer (m)\r\n[K,Kp] = steadyPumpTestLeakyConfined(r,s,Q0,rw,b,bp);\r\nK_correct = 6.959;\r\nKp_correct = 1.126e-5;\r\nassert(abs(K-K_correct)/K_correct\u003c1e-4)\r\nassert(abs(Kp-Kp_correct)/Kp_correct\u003c1e-3)\r\n\r\n%%\r\nfiletext = fileread('steadyPumpTestLeakyConfined.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert') || contains(filetext,'regexp'); \r\nassert(~illegal)\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-11-04T15:23:23.000Z","updated_at":"2026-02-12T15:06:18.000Z","published_at":"2023-11-04T15:23:23.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to determine the hydraulic conductivity \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"K\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eK\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e of a leaky confined aquifer and the hydraulic conductivity \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"K'\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eK\\\\prime\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e of the leaky confining layer given the pumping rate \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Q0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eQ_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and radius \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"rw\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003er_w\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e of the well, the drawdowns \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"s\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e measured at distances \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"r\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003er\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e from the well, and the thicknesses \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"b\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eb\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"b'\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eb\\\\prime\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e of the leaky confined aquifer and the leaking confining layer, respectively. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eBackground\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/59002\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eCody Problem 59002\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e dealt with one-dimensional flow in a leaky confined aquifer. This problem involves flow to a well in a leaky confined aquifer. As in \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/49743\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eother pumping tests\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, the idea is to determine the properties of the aquifer by disturbing it, observing the response, and comparing to an analytical solution. Here the two unknown hydraulic conductivities are determined from two observations of drawdown.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAn analytical solution for the drawdown can be determined by solving the equation\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"d2s/dr2 + (1/r)ds/dr - K's/Kbb' = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\frac{d^2s}{dr^2}+\\\\frac{1}{r}\\\\frac{ds}{dr}-\\\\frac{K\\\\prime s}{\\nKbb\\\\prime} = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewith the boundary conditions that the drawdown far from the well is zero (i.e., \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"s(inf) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es(\\\\infty) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e) and the flow at the well is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Q0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eQ_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Using Darcy’s law and the convention that flow to the well is positive, one finds\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Q0 = -2pi rw b K (ds/dr)|_{r=rw}\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eQ_0 = -2\\\\pi r_wbK\\\\frac{ds}{dr}|_{r=r_w}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe governing equation is related to the one in \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/51783\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eCody Problem 51783\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, and it is the polar coordinates version of the equation in \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/59002\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eCody Problem 59002\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"365\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" 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the incomplete elliptic integrals","description":"Elliptic integrals can be used to evaluate integrals whose integrands have the form R(x,sqrt(P(x))), where R(x,y) is a rational function in x and y and P(x) is a polynomial of degree 4 or less. They appear in calculations of arclength (as in \u003chttps://www.mathworks.com/matlabcentral/cody/problems/45979-compute-the-perimeter-of-an-ellipse Cody Problem 45979\u003e) and analysis of the motion of a pendulum. \r\n\r\nMATLAB provides a function to compute the complete elliptic integrals of the first and second kinds but not the \u003chttps://en.wikipedia.org/wiki/Elliptic_integral _incomplete_ elliptic integrals\u003e of the first, second, and third kinds.\r\n\r\nWrite a function to evaluate the three incomplete elliptic integrals. Follow MATLAB's convention of using the parameter m, which is related to the modulus k through m = k^2.\r\n ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 165.05px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 82.525px; transform-origin: 407px 82.525px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63.05px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.525px; text-align: left; transform-origin: 384px 31.525px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 256.742px 7.79167px; transform-origin: 256.742px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eElliptic integrals can be used to evaluate integrals whose integrands have the form \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"R(x,sqrt(P(x)))\" style=\"width: 79px; height: 20.5px;\" width=\"79\" height=\"20.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24.8917px 7.79167px; transform-origin: 24.8917px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"R(x,y)\" style=\"width: 46px; height: 18.5px;\" width=\"46\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 39.675px 7.79167px; transform-origin: 39.675px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a rational function in \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.79167px; transform-origin: 15.5583px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ey\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.79167px; transform-origin: 15.5583px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"P(x)\" style=\"width: 31.5px; height: 18.5px;\" width=\"31.5\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 254.792px 7.79167px; transform-origin: 254.792px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a polynomial of degree 4 or less. They appear in calculations of arclength (as in\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.79167px; transform-origin: 1.94167px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/45979-compute-the-perimeter-of-an-ellipse\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problem 45979\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 133.808px 7.79167px; transform-origin: 133.808px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e) and analysis of the motion of a pendulum.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 342.542px 7.79167px; transform-origin: 342.542px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eMATLAB provides a function to compute the complete elliptic integrals of the first and second kinds but not the \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Elliptic_integral\"\u003e\u003cspan style=\"perspective-origin: 33.85px 7.79167px; transform-origin: 33.85px 7.79167px; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eincomplete\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"perspective-origin: 1.94167px 7.79167px; transform-origin: 1.94167px 7.79167px; \"\u003e\u003cspan style=\"\"\u003e elliptic integrals\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 110.45px 7.79167px; transform-origin: 110.45px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e of the first, second, and third kinds.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 367.883px 7.79167px; transform-origin: 367.883px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to evaluate the three incomplete elliptic integrals. Follow MATLAB's convention of using the parameter \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003em\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.79167px; transform-origin: 3.88333px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, which is related to the modulus \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ek\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 27.6167px 7.79167px; transform-origin: 27.6167px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e through \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"m = k^2\" style=\"width: 44.5px; height: 19px;\" width=\"44.5\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.79167px; transform-origin: 1.94167px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [F,E,Pi] = ellipticIntegrals(varargin)\r\n% m = 1st argument, phi = 2nd argument, n = 3rd argument (if it's specified)\r\n F = f1(m,phi);\r\n E = f2(m,phi);\r\n Pi = f3(m,phi,n);\r\nend","test_suite":"%%\r\nm = 16/25; phi = pi/7; n = 0.2;\r\nF_correct = 0.458608414805464; E_correct = 0.439360453883539; Pi_correct = 0.464765785383336; \r\n[F,E,Pi] = ellipticIntegrals(m,phi,n);\r\nerrorF = abs(F-F_correct)/F_correct;\r\nerrorE = abs(E-E_correct)/E_correct;\r\nerrorPi = abs(Pi-Pi_correct)/Pi_correct;\r\nassert(all([errorF errorE errorPi] \u003c 1e-8))\r\n\r\n%%\r\nm = 16/25; phi = pi/2; n = 0.2;\r\nF_correct = 1.995302777664729; E_correct = 1.276349943169906; Pi_correct = 2.262478943418680; \r\n[F,E,Pi] = ellipticIntegrals(m,phi,n);\r\nerrorF = abs(F-F_correct)/F_correct;\r\nerrorE = abs(E-E_correct)/E_correct;\r\nerrorPi = abs(Pi-Pi_correct)/Pi_correct;\r\nassert(all([errorF errorE errorPi] \u003c 1e-8))\r\n\r\n%%\r\nm = 1/64; phi = pi/3; n = 0.4;\r\nF_correct = 1.049610464554263; E_correct = 1.044793820490869; Pi_correct = 1.203999963286716; \r\n[F,E,Pi] = ellipticIntegrals(m,phi,n);\r\nerrorF = abs(F-F_correct)/F_correct;\r\nerrorE = abs(E-E_correct)/E_correct;\r\nerrorPi = abs(Pi-Pi_correct)/Pi_correct;\r\nassert(all([errorF errorE errorPi] \u003c 1e-8))\r\n\r\n%%\r\nm = 1/40; phi = pi/3; n = 0.35;\r\nF_correct = 1.051071572767161; E_correct = 1.043347162655471; Pi_correct = 1.182205359771783; \r\n[F,E,Pi] = ellipticIntegrals(m,phi,n);\r\nerrorF = abs(F-F_correct)/F_correct;\r\nerrorE = abs(E-E_correct)/E_correct;\r\nerrorPi = abs(Pi-Pi_correct)/Pi_correct;\r\nassert(all([errorF errorE errorPi] \u003c 1e-8))\r\n\r\n%%\r\nm = 0; phi = rand(1); n = 0.4;\r\nF_correct = phi; E_correct = phi; \r\n[F,E,~] = ellipticIntegrals(m,phi,n);\r\nerrorF = abs(F-F_correct)/F_correct;\r\nerrorE = abs(E-E_correct)/E_correct;\r\nassert(all([errorF errorE] \u003c 1e-8))","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2022-06-01T22:25:14.000Z","deleted_by":null,"deleted_at":null,"solvers_count":15,"test_suite_updated_at":"2020-12-25T06:02:12.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2020-06-22T22:58:33.000Z","updated_at":"2026-01-09T12:16:35.000Z","published_at":"2020-06-23T04:29:36.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eElliptic integrals can be used to evaluate integrals whose integrands have the form \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"R(x,sqrt(P(x)))\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eR(x,\\\\sqrt{P(x)})\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"R(x,y)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eR(x,y)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a rational function in \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"P(x)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eP(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a polynomial of degree 4 or less. They appear in calculations of arclength (as in\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/45979-compute-the-perimeter-of-an-ellipse\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem 45979\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e) and analysis of the motion of a pendulum.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eMATLAB provides a function to compute the complete elliptic integrals of the first and second kinds but not the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Elliptic_integral\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eincomplete\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e elliptic integrals\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e of the first, second, and third kinds.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to evaluate the three incomplete elliptic integrals. Follow MATLAB's convention of using the parameter \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"m\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, which is related to the modulus \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"k\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e through \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"m = k^2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003em = k^2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":45988,"title":"Evaluate the zeta function for real arguments \u003e 1","description":"The \u003chttps://en.wikipedia.org/wiki/Riemann_zeta_function Riemann zeta function\u003e is important in number theory. In particular, the \u003chttps://en.wikipedia.org/wiki/Riemann_hypothesis Riemann hypothesis\u003e, one of the seven \u003chttps://en.wikipedia.org/wiki/Millennium_Prize_Problems Millenium Prize Problems\u003e, states that the non-trivial zeros of the zeta function all have real part equal to 1/2. The truth of the Riemann hypothesis has consequences for the distribution of prime numbers. \r\n\r\nThis problem deals only with values of the zeta function for real arguments greater than 1. For a positive integer argument x, the zeta function is the sum of the reciprocals of integers raised to the power of x. Euler showed that when x is an even integer, the value of the zeta function is proportional to pi^x, and \u003chttps://www.mathworks.com/matlabcentral/cody/problems/45939-estimate-pi-from-certain-values-of-the-zeta-function Cody Problem 45939\u003e uses this fact to estimate pi. Less is known about the zeta function for odd integer arguments, but Apery provided that zeta(3), now known as \u003chttps://en.wikipedia.org/wiki/Apéry's_constant Apery's constant\u003e, is irrational. \r\n\r\nEvaluate the zeta function for real arguments greater than 1. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 207px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 103.5px; transform-origin: 407px 103.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12.0667px 7.8px; transform-origin: 12.0667px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Riemann_zeta_function\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eRiemann zeta function\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 145.733px 7.8px; transform-origin: 145.733px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is important in number theory. In particular, the\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Riemann_hypothesis\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eRiemann hypothesis\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 55.6167px 7.8px; transform-origin: 55.6167px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, one of the seven\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Millennium_Prize_Problems\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eMillenium Prize Problems\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 335.65px 7.8px; transform-origin: 335.65px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, states that the non-trivial zeros of the zeta function all have real part equal to 1/2. The truth of the Riemann hypothesis has consequences for the distribution of prime numbers.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 105px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 52.5px; text-align: left; transform-origin: 384px 52.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 378.467px 7.8px; transform-origin: 378.467px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThis problem deals only with values of the zeta function for real arguments greater than 1. For a positive integer argument \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABwAAAAkCAYAAACaJFpUAAAA60lEQVRIie2U0Q2DIBBA3w5s4AIu4ASdgA3cwA1cgRkcwR26gjOwgv3gLhKi0kT0o+UlJuIFH3B3QKVSqVR+kgawwCDvaWyQ+GVewATMwCrPFMVt9H3dWcwlVOoBA3TAIotyyUKK0LPtRGVdaUlMGwk9hXKWw4tweUIGIU8r8H5CZkR0S0Xu4RLhrTm0hLwZtjy6u2StyFoZax61cPSom2jsCH07yvgUIz+dCL3nCc2txP1oRWYP4qtIT+mSCX0SbzLx9LrzOaFOGji+SbpMXOfPPNRCysgXR1oKLbZs0ZSU3X45KJaHdlb5Ez4sHUr70Yy5uAAAAABJRU5ErkJggg==\" alt=\"x\" style=\"width: 14px; height: 18px;\" width=\"14\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 248.15px 7.8px; transform-origin: 248.15px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e the zeta function is the sum of the reciprocals of integers raised to the power of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 80.5167px 7.8px; transform-origin: 80.5167px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Euler showed that when \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 35.7833px 7.8px; transform-origin: 35.7833px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is an even integer, the value of the zeta function is proportional to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACIAAAAmCAYAAACh1knUAAABLklEQVRYhe2WXa3EIBBGjwccYAADq6AK6qAO6mAtVMNKqIdauBqwwH1gJkAvm93tpjQ34SQ80D8+5psZCp1Op9P5CAfcsrmRuWklwAITsAFBFnfAj8znVkKUSRa+A6sIvAQnQgIwXCVCCURLLsWQ8uIrWywx0V6NZ4s8SAk7HhEwknbyznCVbwwi5CbPLMQIvV0xywcC9v5bYnXMxEgYGQHw8u28rzzlLruwMvYLjaRyrGHk/YXSrokUnZcYeUGZ+dt8VlKDaoan9F8j5FuKUAtyW7RLLi2FqAV5LmgVHSrDI6gFeS7krbrZyanl6yvXtuzawImHmOZBoKwg7Y4qxHFS0lrKTuopLcgbmObPKbmSR6J2ZG+7+48zREDcvXbV2n+Dk3srDaum0+l0/hW/ktdyQqlGSvEAAAAASUVORK5CYII=\" alt=\"pi^x\" style=\"width: 17px; height: 19px;\" width=\"17\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5667px 7.8px; transform-origin: 15.5667px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/45939-estimate-pi-from-certain-values-of-the-zeta-function\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problem 45939\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 80.8833px 7.8px; transform-origin: 80.8833px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e uses this fact to estimate \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eπ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 27.6167px 7.8px; transform-origin: 27.6167px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Less is known about the zeta function for odd integer arguments, but Apery proved that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"zeta(3)\" style=\"width: 30.5px; height: 18.5px;\" width=\"30.5\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 48.2333px 7.8px; transform-origin: 48.2333px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, now known as\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Ap%C3%A9ry's_constant\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eApery's constant\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.8833px 7.8px; transform-origin: 10.8833px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, is irrational.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 186.7px 7.8px; transform-origin: 186.7px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eEvaluate the zeta function for real arguments greater than 1.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function z = zeta1(x)\r\n z = f(x);\r\nend","test_suite":"%%\r\nx = 3/2;\r\nz_correct = 2.612375348685488;\r\nassert(abs(zeta1(x)-z_correct)/z_correct \u003c 1e-8)\r\n\r\n%% \r\nx = 2;\r\nz_correct = pi^2/6;\r\nassert(abs(zeta1(x)-z_correct)/z_correct \u003c 1e-8)\r\n\r\n%%\r\nx = 3;\r\nz_correct = 1.202056903159594;\r\nassert(abs(zeta1(x)-z_correct)/z_correct \u003c 1e-8)\r\n\r\n%%\r\nx = 4;\r\nz_correct = pi^4/90;\r\nassert(abs(zeta1(x)-z_correct)/z_correct \u003c 1e-8)\r\n\r\n%%\r\nx = 5;\r\nz_correct = 1.036927755143370;\r\nassert(abs(zeta1(x)-z_correct)/z_correct \u003c 1e-8)\r\n\r\n%%\r\nB = [1/6 -1/30 1/42 -1/30 5/66 -691/2730 7/6 -3617/510 43867/798 -174611/330];\r\nn = randi(10);\r\nx = 2*n;\r\nz_correct = (-1)^(n+1)*B(n)*(2*pi)^(2*n)/(2*factorial(2*n));\r\nassert(abs(zeta1(x)-z_correct)/z_correct \u003c 1e-8)","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":17,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-06-24T20:47:20.000Z","updated_at":"2026-01-09T12:24:36.000Z","published_at":"2020-06-24T21:36:21.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Riemann_zeta_function\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eRiemann zeta function\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e is important in number theory. In particular, the\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Riemann_hypothesis\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eRiemann hypothesis\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, one of the seven\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Millennium_Prize_Problems\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eMillenium Prize Problems\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, states that the non-trivial zeros of the zeta function all have real part equal to 1/2. The truth of the Riemann hypothesis has consequences for the distribution of prime numbers.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis problem deals only with values of the zeta function for real arguments greater than 1. For a positive integer argument \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex,\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e the zeta function is the sum of the reciprocals of integers raised to the power of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Euler showed that when \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is an even integer, the value of the zeta function is proportional to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"pi^x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi^x\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/45939-estimate-pi-from-certain-values-of-the-zeta-function\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem 45939\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e uses this fact to estimate \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Less is known about the zeta function for odd integer arguments, but Apery proved that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"zeta(3)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\zeta(3)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, now known as\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Apéry's_constant\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eApery's constant\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, is irrational.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEvaluate the zeta function for real arguments greater than 1.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":57620,"title":"Compute frequency factors for the normal distribution","description":"In frequency analysis in hydrology, the streamflow corresponding to a specified exceedance probability (or return period ) can be computed as\r\n\r\nwhere and are the mean and standard deviation of the streamflow series, respectively, and is the frequency factor. \r\nWrite a function to compute the frequency factor for the normal distribution given the exceedance probability as a vector. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 135px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 67.5px; transform-origin: 407px 67.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 43px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.5px; text-align: left; transform-origin: 384px 21.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 156.242px 8px; transform-origin: 156.242px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn frequency analysis in hydrology, the streamflow \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"QT\" style=\"width: 20.5px; height: 20px;\" width=\"20.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 164.95px 8px; transform-origin: 164.95px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e corresponding to a specified exceedance probability \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ep\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.6667px 8px; transform-origin: 32.6667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e (or return period \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"T = 1/p\" style=\"width: 55.5px; height: 18.5px;\" width=\"55.5\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 67.2917px 8px; transform-origin: 67.2917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e) can be computed as\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 11px; text-align: left; transform-origin: 384px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"QT = mu + KT sigma\" style=\"width: 89.5px; height: 20px;\" width=\"89.5\" height=\"20\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 11px; text-align: left; transform-origin: 384px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 8px; transform-origin: 21.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eμ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eσ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 249.592px 8px; transform-origin: 249.592px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are the mean and standard deviation of the streamflow series, respectively, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"KT\" style=\"width: 19.5px; height: 20px;\" width=\"19.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 74.275px 8px; transform-origin: 74.275px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the frequency factor. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 373.275px 8px; transform-origin: 373.275px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to compute the frequency factor for the normal distribution given the exceedance probability as a vector. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function KT = normFreqFactor(p)\r\n KT = trapz(QT:inf,exp(-Q.^2));\r\nend","test_suite":"%%\r\np = 0.001;\r\nKT_correct = 3.090;\r\nassert(abs(normFreqFactor(p)-KT_correct)\u003c1e-3)\r\n\r\n%%\r\np = 0.002;\r\nKT_correct = 2.878;\r\nassert(abs(normFreqFactor(p)-KT_correct)\u003c1e-3)\r\n\r\n%%\r\np = 0.01;\r\nKT_correct = 2.326;\r\nassert(abs(normFreqFactor(p)-KT_correct)\u003c1e-3)\r\n\r\n%%\r\np = 0.02;\r\nKT_correct = 2.054;\r\nassert(abs(normFreqFactor(p)-KT_correct)\u003c1e-3)\r\n\r\n%%\r\np = 0.04;\r\nKT_correct = 1.751;\r\nassert(abs(normFreqFactor(p)-KT_correct)\u003c1e-3)\r\n\r\n%%\r\np = 0.1;\r\nKT_correct = 1.282;\r\nassert(abs(normFreqFactor(p)-KT_correct)\u003c1e-3)\r\n\r\n%%\r\np = 0.2;\r\nKT_correct = 0.842;\r\nassert(abs(normFreqFactor(p)-KT_correct)\u003c1e-3)\r\n\r\n%%\r\np = 0.3;\r\nKT_correct = 0.524;\r\nassert(abs(normFreqFactor(p)-KT_correct)\u003c1e-3)\r\n\r\n%%\r\np = 0.5;\r\nKT_correct = 0;\r\nassert(abs(normFreqFactor(p)-KT_correct)\u003c1e-3)\r\n\r\n%%\r\np = 0.8;\r\nKT_correct = -0.842;\r\nassert(abs(normFreqFactor(p)-KT_correct)\u003c1e-3)\r\n\r\n%%\r\nT = [5 10 20 25 50 100 250 500 1000];\r\nKT_correct = [0.842 1.282 1.645 1.751 2.054 2.326 2.652 2.878 3.090];\r\nassert(all(abs(normFreqFactor(1./T)-KT_correct)\u003c1e-3))\r\n\r\n%%\r\np = rand(1,randi(15));\r\nK1 = normFreqFactor(p);\r\nK2 = normFreqFactor(1-p);\r\nassert(all(abs(K1+K2)\u003c1e-3))\r\n\r\n%%\r\nfiletext = fileread('normFreqFactor.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert') || contains(filetext, 'regexp') || contains(filetext, 'interp'); \r\nassert(~illegal)","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2023-01-29T19:23:19.000Z","deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-01-29T19:23:11.000Z","updated_at":"2026-01-04T12:13:24.000Z","published_at":"2023-01-29T19:23:20.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn frequency analysis in hydrology, the streamflow \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"QT\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eQ_T\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e corresponding to a specified exceedance probability \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e (or return period \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"T = 1/p\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eT = 1/p\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e) can be computed as\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"QT = mu + KT sigma\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eQ_T = \\\\mu + K_T \\\\sigma\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"mu\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\mu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"sigma\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\sigma\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e are the mean and standard deviation of the streamflow series, respectively, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"KT\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eK_T\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the frequency factor. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to compute the frequency factor for the normal distribution given the exceedance probability as a vector. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51322,"title":"Solve an ODE: diffusion problem 1","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 264.25px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 132.125px; transform-origin: 407px 132.125px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28px 7.91667px; transform-origin: 28px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 295.017px 7.91667px; transform-origin: 295.017px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn the solution of a problem involving diffusion, the following ordinary differential equation arises\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f''+a eta f' = 0\" style=\"width: 91px; height: 18px;\" width=\"91\" height=\"18\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.25px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.125px; text-align: left; transform-origin: 384px 21.125px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 7.91667px; transform-origin: 21.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 298.233px 7.91667px; transform-origin: 298.233px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a positive constant and primes denote differentiation with respect to the independent variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 45.1167px 7.91667px; transform-origin: 45.1167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The problem has the conditions \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f(0) = f0\" style=\"width: 60px; height: 20px;\" width=\"60\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIIAAAAmCAYAAAAIjkMFAAADx0lEQVR4nO2babGrQBCFjwccxAAGogAFOIiDOIiFaEACHmIBDVjI/TFzioYHTPew581XlbobMNuZ3pgLJBKJRCKRSCQSO3ADcDfekwF4+69HUQB4HNj+z5ADaADUcIvaQjexmb8n365rap5wff8Vcrg1ePqvt60bvAP4Aqj8zw//c4v5XZ4B+MBuQbbkDeB1dCcWksNtrhbO0t3hxPCFG9smljfzDbboFFf4RuvAvRXOOekN3BiuSA63Fl/8awEohhobiIEPr0IXDigRthhHccd5+xbig27nj9H4v68eD1F9pfG+Bk5EZ+WD68ULJdxafDHtbl/o3PZqMYNs2PJQ3nfmHXeFPg6hNZizZnTbMZt3kso/sDHe90E4fjiaDBuZ0I24oVvgj/I6qzvvwXTkic4tNOJ3D8zvInYkxi0Uop0i0A7QRctPOPVbd/cVBEuYuWkWmNe1SxrkxL4HDcsJn4Mm15IyFuhEJz9T0T1rGsPr24nrp+AYYyjRzcmSjxYG7RYhWF16sGHLovI+bQGJwql9O6X/Xg5GLq5Mn6Y+WjFY+yoZ9jHmY9mxcj1CQa6cn8WFPMYH1jSLE6Th5p8/pnA5cC5uhs4SvNAN8ja4vlH2mYWxmILXGhbBEp/I8YXukyJdLASqyupDLULg4KbMF1MhuRPmdrzMcjRWgX73CgGjFEKoSCfd5iIhyMjTWhm0CIFlUs3zrLtB438phDPXOwit164xgsxFraVYixA0fpLuQxuvcMI0qdOVhLB71gD0TbI1JbMKQaNauRs+gT5xwjRVw5gMR967Z9bAukfIXWuvU8HFnCtcTEERaQREXxaakGHgOOeuLH6fz40Rwt5ZA9CvLE4hLcdiS6dNU8awROIyM5kKarhreW3IZbF9jRAp2hj2zhrYZshFLrHmPXLxoJhaNe/XDFIOrEG/OligX+LO0HcRY2LgK3OtgGvEWb0j0b59XGwN5GTHRpwN9IvBgc2ZT9mP9+DvLzhBlL7dUAxB6EvPeF5iDhbUxqwo126V8wicaOuLJskLev/Ho2xjIqgxLsYHxquL1cT1YzAz2vx41wawxE4rKk8oVVjpjWrI9GhgHcLiWuT5uxLhQkiGzk9rrh9S4TovnKYYvnRbTdQy9VhamnzhvBNNoZ7hQO0pYfC2RgDF9wKrHY5YkRrXiw02g/9rIKN7RulrnTqmLzuTH37gepnCpsgiTYYu7Vv7HF+BjU7VRpBDn1X8NzANofm2pHwxbR0thhwrRtS/xg3OVO7xXzJHL8CZ3FMikUgkEolEInE+/gCQzs5E34QVuwAAAABJRU5ErkJggg==\" alt=\"f(infinity) = 0\" style=\"width: 65px; height: 19px;\" width=\"65\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 194.342px 7.91667px; transform-origin: 194.342px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve this problem—that is, return values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 69.2333px 7.91667px; transform-origin: 69.2333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.8333px 7.91667px; transform-origin: 40.8333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eBackground\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 372.133px 7.91667px; transform-origin: 372.133px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe physical problem involves diffusion of a quantity from one point where the concentration is maintained at a constant value into a medium in which the concentration is zero far away. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function f = diffusion1ODE(eta,a,f0)\r\n% eta = independent variable\r\n% a = constant\r\n% f0 = value of f at eta = 0\r\n\r\n f = g(eta,a,f0)\r\nend","test_suite":"%%\r\na = 1/2; \r\nf0 = 1;\r\neta = 0:0.2:1;\r\nf_correct = [1 0.887537083981715 0.777297410789522 0.671373240540873 0.571607644953331 0.479500122186953];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 2; \r\nf0 = 1;\r\neta = 0.5;\r\nf_correct = 0.479500122186953;\r\nassert(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10)\r\n\r\n%%\r\na = 1;\r\nf0 = 0.5;\r\neta = [0.12 0.23 0.456 0.789 1.011];\r\nf_correct = [0.452241573979416 0.409045884857994 0.324194989107724 0.215056003266342 0.156008214896009];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 2;\r\nf0 = 1;\r\neta = 1:4;\r\nf_correct = [0.157299207050285 0.004677734981047 2.209049699858544e-05 1.541725790028002e-08];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 3/4;\r\nf0 = 4/3;\r\neta = 3*logspace(-2,0,3);\r\nf_correct = [1.305696910508165 1.060016229285651 0.012499691279247];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 0.01;\r\nf0 = 1;\r\neta = rand/120;\r\nf_correct = polyval(flip([1 -0.0797885 0 0.000132981 0 -1.99471e-7]),eta);\r\nf = diffusion1ODE(eta,a,f0);\r\nassert(all(abs(f-f_correct)\u003c1e-7))","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":6,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-04-08T01:02:26.000Z","updated_at":"2025-05-04T20:55:44.000Z","published_at":"2021-04-08T01:09:52.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn the solution of a problem involving diffusion, the following ordinary differential equation arises\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f''+a eta f' = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime\\\\prime + a\\\\eta f\\\\prime = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a positive constant and primes denote differentiation with respect to the independent variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The problem has the conditions \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(0) = f0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(0) = f_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(infinity) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(\\\\infty ) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve this problem—that is, return values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eBackground\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe physical problem involves diffusion of a quantity from one point where the concentration is maintained at a constant value into a medium in which the concentration is zero far away. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":54710,"title":"Compute the period of a pendulum started from a finite initial angle","description":"Cody Problem 49830 asks for the period of a pendulum swinging through a small angle. Here the pendulum started at rest from an angle that is not necessarily small. The other assumptions are similar (no friction or drag, massless rod). \r\nWrite a function that takes the initial angle and returns , where is the length of the pendulum and is the acceleration of gravity. In the limit as the initial angle approaches zero, the function should produce .","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 94.1px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 47.05px; transform-origin: 407px 47.05px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 43px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.5px; text-align: left; transform-origin: 384px 21.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/49830\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003eCody Problem 49830\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 61.45px 8px; transform-origin: 61.45px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e asks for the period \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eT\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 240.792px 8px; transform-origin: 240.792px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e of a pendulum swinging through a small angle. Here the pendulum started at rest from an angle \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"theta0\" style=\"width: 15px; height: 20px;\" width=\"15\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 306.883px 8px; transform-origin: 306.883px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e that is not necessarily small. The other assumptions are similar (no friction or drag, massless rod). \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.1px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.05px; text-align: left; transform-origin: 384px 21.05px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 168.683px 8px; transform-origin: 168.683px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that takes the initial angle and returns \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"T\\sqrt{g/L}\" style=\"width: 52.5px; height: 20.5px;\" width=\"52.5\" height=\"20.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24.8917px 8px; transform-origin: 24.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eL\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 107.358px 8px; transform-origin: 107.358px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the length of the pendulum and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eg\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 20.6083px 8px; transform-origin: 20.6083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the acceleration of gravity. In the limit as the initial angle approaches zero, the function should produce \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"2pi\" style=\"width: 19.5px; height: 18px;\" width=\"19.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function T = pendulumPeriod(theta0)\r\n T = theta0-theta0^3/3!+theta0^5/5!+higher order terms;\r\nend","test_suite":"%%\r\nth = pi/7;\r\nT_correct = 6.363207946270837;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%%\r\nth = pi/5;\r\nT_correct = 6.44181661515865;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%% Problem 1 on p. 194 of Davis (1962)\r\nth = pi/4;\r\nT_correct = 6.534345229832591;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%%\r\nth = pi/3;\r\nT_correct = 6.743001419250384;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%%\r\nth = pi/2;\r\nT_correct = 7.416298709205487;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%%\r\nth = 36*pi/37;\r\nT_correct = 18.190113206504414;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%% \r\nth = 72*pi/73;\r\nT_correct = 20.902949604823448;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%% \r\nth = 0;\r\nT_correct = 2*pi;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%%\r\nfiletext = fileread('pendulumPeriod.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert') || contains(filetext, 'switch'); \r\nassert(~illegal)\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":3,"created_by":46909,"edited_by":46909,"edited_at":"2022-06-06T01:13:14.000Z","deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2022-06-06T00:37:45.000Z","updated_at":"2026-01-09T20:11:24.000Z","published_at":"2022-06-06T01:13:14.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/49830\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eCody Problem 49830\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e asks for the period \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"T\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eT\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e of a pendulum swinging through a small angle. Here the pendulum started at rest from an angle \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"theta0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\theta_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e that is not necessarily small. The other assumptions are similar (no friction or drag, massless rod). \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that takes the initial angle and returns \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"T\\\\sqrt{g/L}\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eT\\\\sqrt{g/L}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"L\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eL\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the length of the pendulum and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"g\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eg\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the acceleration of gravity. In the limit as the initial angle approaches zero, the function should produce \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"2pi\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e2\\\\pi\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51975,"title":"Compute a sum of Ramanujan","description":"Srinivasa Ramanujan defined the following function:\r\n\r\nWrite a function to compute for various values of . See also Cody Problems 45960 and 46000.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 105px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 52.5px; transform-origin: 407px 52.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 160.675px 7.79167px; transform-origin: 160.675px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSrinivasa Ramanujan defined the following function:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 45px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22.5px; text-align: left; transform-origin: 384px 22.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAUUAAABaCAYAAAAvg0SnAAALrUlEQVR4nO2dzZGruhaFVw7OwAk4AQ/u2BE4A2fQGTiFjsEhOIeO4FY5gjfoFO4b4FVsqyUhQBIC1lelOtWmaTgYFlv7T4AQQgghhBBCCCGEEEIIIYQQQgghhBBCCCGEEEIIIUQGzu8hhBC75gbgF8B/7/GDT3E8vz/7D8ALwBeAQ+VzFEKIKlzQid0JndB9v3/+BXB8f86fn+jF8xsSRiHEBnmhsxQtD3TC93xvf5htB/RWo6baQohNcUQnbhfn8wM6MaQwutC6/Cp6dkIIUZkzOnG7erZR+F7wT5P/A3Avd2pCCFEf+gsf+Ct8N/SBF5/4hcRUCCFWDafJN3TCeEA3Leb0+NdsJ9f350IIsTk4TXYHRfBqPvtGZzX+orMyhRBik5zRCd7z/a8bVT45249Vz04IIYQQQgghhBBCCCH2wQF/q4WEEGJ3MK3pF0pdEmJRmF/YytijILgdh/Z4DYRoBl/O4S86scw57ujSc56e47ljT1UvN3TpTLZ2XKIoxMKw9ZcdpRs3nNCJH7vr2PFT+NitwheGRFGIhbGtveyo1ebL+tNqH7slJIpCNAQbPbjT6JpNYQ/oLcfvisdtBYmiEI1hu9xwPKJ7lD2PvZUGShSFaBCfj2+JvLk79teQVqIoRIPYKKgdS3S42VvARaIoRKOwu7YdoW7apc9jT1NoiWJhTugucq4b+fj+e+qJtw98Sd17DH7URKI4ABNdp/hzcgsi4dRqi8K41v/TAf2C9LmtKl+i9Z4Sq2sjUYxgqwzG3oQndMJVatrBv7+V9XvP6NcmXhN88fmmubly/I74zB3kA7unKW1NJIoR7uhvwjHiwyTc0tHCL/iXsVwTrqis6UYMLQdgh7sEac5j7S0AUguJYgRWF4y9+b7QWQo18C2IvhZu6K7VBb0ltJYb8YDuXJ/4tAgv+Gs55rLm7UuaQ0uV5keiGICLjY+tP+XDUsvnw84ea59Gl74RGcnN9ffv6F6WoetuhTHXSytUBpjLGhUdEsUAdiW0Mb4h1o7W4oBtON7XJopDq+LZ6W7OapQT/P7Ftb8UW0KiGIAVBWMvzAv1S7KeWL9/aU2ieERaWkypEj37wuZYu2+5JWiNSxTR3exnfHYKeb1/PmM4XYTF/HOmSyf0fd3IUJoHc9nWbC2sSRRTKdn+y1cGuLdyvNwc8Tcv9Avrfq4mc0R/kz3wWZDPQADfHneELxL3G5uKYVs2PdBHla1jPWWqNiUFhC+CuWPujbM1UbQ+6RK5l6EywD22+soBjaHQ2JXf9oreKuTNa98WtNDsTR6K+N2dfVJgvqH78NBXmPIg84GfYinkaoM/98HfmihyiltyWutrM7al3FWxAKFgCq1CN63mifiNx+2pWKe5T1R4bkP+Kwr2FFG8oq/amTPmJhJvTRT5/yldoeN7qS3RZkxsAFslYMXEWmiuRWjTLOaKohVEX+TYnkdKZHntzvYcohib3tO18TvwezlEbI7lPgU3L1K10WISoYcwlopjUyFifzOFocRwm9KRMh2SKOZxA8y1sujrqylMzI2N3U9CRLEtmdxIsV08yGJ9iiHxSRVFK7whBy7PI/UmX/u0KYcoxqb3Nuk59ntzrbsHyjQBiUHfYiyZXIgoNp3B9YUx6OEKnxuR9pEqijxGqBQwJahj4UMx5YFW9Dkf36gviHQDDSWTCxHFRuosNprnWpA2/SF009PpHQs62GOEBM9aqykpFnzgp+RHKvqchyUE0Zb9KRVHTMZOnV2/j7UGT4F9YtYYp8WxG9Qewzd1drstpzxkKceN7avo8zy4wHzsu8phTbvwmi1d4vkvgP9pND/+CXx/H6ITii67FmRqt5yU1BhrmbmiyIg0p/c83hVxK5D5kWtmraJ4xXDfzBPyB0A4m2ihS45EcR3jn8D3B8BvKR4Cn9OyS02K/UE84HGBX5TP6P1CL7OdD13s2EvUW+dmjaJIC51VSL5xfx8zZ3oO70ml3ohs0LKyb3grVpyO2JsvderDfYZEjMf6eQ/rKOe2XwwLIq1T1+pcE67LoIR/LLco+poyhEbOLtk8riLNIjusL/5F7yS3Fhxrj8fezMwZi013D+YYD/z1Cd3QrwszdOPfUa+hbW7O6K+9O+7IK/S5u+SM8bfmmuLSvaJSvrqwScvuovuhVJwp3FDnxqUAK/I4DDugrLWTjFJv6mNdWdZg2sULyabJ5HpociQCD/GA/Ep7YOnUm6HWdVvE7aB/RP8dLB3tr0IoFWcOvJFL3cRXyK+0F5iNsMTDyPt4Dle08fK+oLuWKc+Mr10Y4w4tRPyLUyryyZXqck93Sq0nLdqDqTdLico35olAaUHkets3pAneGdONCfqlWwxq2gbZWa53KJk7Bwekv51SYCt8CeL2sSk/S8Cc2qkvdc5mSnBA9xzYmvbUY7FgYSzfaC/17YTunKzvc7Yle0F3MZ9QwEK0A6dqS7hI+CIfIzQujJSXDgqN7RNAxvr82Rm/RSsRSGsyI8RqoaDUXqWPEXrbIm/qekNP1Jny2xzXMUYNxTRl3aVvfF6TFkXHVsoJsSkY9axhZdEfx5xdX+L5HN9bjYj1HDFgm7cUOF2fYz2XhN/fmnuqCvEHm3rD6V2u4SaUp1TiTPWfsSihBnPEgC6KMeLN47Xm08+dUihEE/iWLV1yTJkmclo6Z5nfMcwVg7G+yBaXE57qQhCiaWyubAtjanoa/x9zpv5sXEw4zXeFyPYrmCoG7MqeytfI358KU2ysFXuC/7rapZCFEI1Ba3csTDWj5Ubx4Wc+kb57tl3xaXF/IS7QobZ7B/in1S+UtYLZFesH3bk/0HdaClngdLlYl8WX+ZyVOXurShKiCX4wvkHJFf5GF9YS9PkofWIAdIIw1OPS/q7P0qTvkE1hGGgpFVE/mGO6wmdzMV2Bty0P3f3o221pqi/Eajghz8MzNujB/DpfAxXrK3NLHENiQEtx7PFdUTyjj04z6FXK0rIBNp8lGGp+DXy+OKxgfgf+lhCrh41qS+X9sQ1aruDBGFG0Ccc+wblEttt9KQY3jA+41F6j28UKYuj7jW133Qu0aBVwEZuGfq/cDSHu+HTS1xRFtkKLRX95br7AhhUDCsEUYVtaFIeuv7WIfdkALO97oO/kI0EUm4eWXKnp2xKiaFesDB3XLssR2vZAfOo5xJyVMGnBTxmcjvMahF4M1iJ2r5O7P18y6rcpNk/IWsr592uKon2YQ1PGWC2v7X9KazHklxyCKURTrKtQ1/iUwe5BMfcB0Iu/7/u3aVy2DLG1phVCZIWWTMkefjlF8Ynh6PPQMr/AZ9eX2P5nfE5Bxyacz+0ENAcKWeiFZ18Mvuk9U4/oQrApOEq/EZuFD+0F3Y1v1+85ol8/JHX4HpacokjrJ4bNI/Qd09Yz+wSD+1N8reU5ttwv5XxLYK1d3wuPNfAxsed2WttDIirEJuDbn+LxwOcDklrHHHu4copiKMXFYs/ZPSZ7MLr+xLv5m64YAJ9CS6vvgOGIPdsG1ia2/jytPvflwY49wKeo2gCcz5XAJrxCrB5GHtlT0QpDTnKKIs859hCGprtc8O3kbLfNXc+Bfe3n7ErvWynTUrtO22KjyraLPiPIN3x25XH7U1pr2n5v9nMuspWayC5E8zBPjxHWUqkWOUUR6M83hJ3uUhRY2sY6ZxtAcMvXQudrxfaF4YADrdqlKj5soOWF3oKmkFuL2m0lx22+62wrYEqXJQpRFVvbG/I9teZTBHpRiwUvOBVkUror+LQOXUvPrsnug23SUnxqrK1eEvqI6Se238EZ4fXnbRTbhW6DJ9psiCvEZF7oI4k2ymhpzadIHmg7NeSM+p3NhRAzoBAyCMCpFi2gnNUtJUSR0dMWqyvoZ9vF+s1CbAX6uyiC1lr8Qr4H2vr3cjvjT5iWUF2akt1uhBCFoK/J+uUuCPuRpsDILv1TJaLbPOdWhLH0OtRCCDGIza9bkguU1CyEEEIIIYQQQgghhBBCCCGEEEJk4v/WSOigWbq8pQAAAABJRU5ErkJggg==\" alt=\"phi(a) = 1+2 Sum[1/((a k)^3 - a k),{k,1,infinity}]\" style=\"width: 162.5px; height: 45px;\" width=\"162.5\" height=\"45\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 86.9917px 7.79167px; transform-origin: 86.9917px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to compute \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"phi(a)\" style=\"width: 32.5px; height: 18.5px;\" width=\"32.5\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 66.5083px 7.79167px; transform-origin: 66.5083px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e for various values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 82.85px 7.79167px; transform-origin: 82.85px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. See also Cody Problems \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/45960\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration: underline; text-decoration-line: underline; \"\u003e45960\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.79167px; transform-origin: 15.5583px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46000-compute-the-harmonic-numbers\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration: underline; text-decoration-line: underline; \"\u003e46000\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.79167px; transform-origin: 1.94167px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = Ramanujanphi(a)\r\n y = f(a);\r\nend","test_suite":"%%\r\na = 2;\r\ny_correct = 2*log(2);\r\nassert(abs(Ramanujanphi(a)-y_correct)\u003c1e-14)\r\n\r\n%%\r\na = 3;\r\ny_correct = log(3);\r\nassert(abs(Ramanujanphi(a)-y_correct)\u003c1e-14)\r\n\r\n%%\r\na = 4;\r\ny_correct = (3/2)*log(2);\r\nassert(abs(Ramanujanphi(a)-y_correct)\u003c1e-14)\r\n\r\n%%\r\na = 5;\r\nphi = (1+sqrt(5))/2;\r\ny_correct = (sqrt(5)/5)*log(phi)+(1/2)*log(5);\r\nassert(abs(Ramanujanphi(a)-y_correct)\u003c1e-14)\r\n\r\n%%\r\na = 6;\r\ny_correct = (1/2)*log(3)+(2/3)*log(2);\r\nassert(abs(Ramanujanphi(a)-y_correct)\u003c1e-14)\r\n\r\n%%\r\na = 7;\r\ny_correct = (2/7)*(log(14)+2*cos(pi/7)*log(cos(pi/14))+2*log(cos(3*pi/14))*sin(pi/14)-2*log(sin(pi/7))*sin(3*pi/14));\r\nassert(abs(Ramanujanphi(a)-y_correct)\u003c1e-14)\r\n\r\n%%\r\na = 8;\r\ny_correct = log(2)+(sqrt(2)/8)*log((2+sqrt(2))/(2-sqrt(2)));\r\nassert(abs(Ramanujanphi(a)-y_correct)\u003c1e-14)\r\n\r\n%%\r\na = 12;\r\ny_correct = (1/2)*log(2)+(1/4)*log(3)+(sqrt(3)/6)*log((sqrt(3)+1)/(sqrt(3)-1));\r\nassert(abs(Ramanujanphi(a)-y_correct)\u003c1e-14)\r\n\r\n%%\r\na = 18;\r\nb = pi/9;\r\ny_correct = (1/9)*(log(2)+2*log(6)+log(sqrt(3)/2)+2*cos(b)*log(cos(b/2))+2*cos(2*b)*log(cos(b))-2*cos(b)*log(sin(b/2))-2*cos(2*b)*log(sin(b))+2*log(cos(2*b))*sin(b/2)-2*log(sin(2*b))*sin(b/2));\r\nassert(abs(Ramanujanphi(a)-y_correct)\u003c1e-14)\r\n\r\n%%\r\na = [11 13 17];\r\nsum_correct = 3.003409919427940;\r\nassert(abs(sum(Ramanujanphi(a))-sum_correct)\u003c1e-14)\r\n\r\n%%\r\na = [36 54 72 100];\r\ny_correct = [1.0000515628258977 1.0000152722224909 1.0000064421348023 1.000002404321212];\r\nk = randi(4);\r\nassert(abs(Ramanujanphi(a(k))-y_correct(k))\u003c1e-14)","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":17,"test_suite_updated_at":"2021-06-05T14:29:40.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-06-05T14:22:18.000Z","updated_at":"2026-01-20T21:12:40.000Z","published_at":"2021-06-05T14:23:20.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSrinivasa Ramanujan defined the following function:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"phi(a) = 1+2 Sum[1/((a k)^3 - a k),{k,1,infinity}]\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\phi(a) = 1+2\\\\sum_{k=1}^\\\\infty\\\\frac{1}{(a k)^3 – a k}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to compute \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"phi(a)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\phi(a)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for various values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. See also Cody Problems \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/45960\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e45960\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46000-compute-the-harmonic-numbers\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e46000\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":52125,"title":"Compute the sum of reciprocals of quadratics","description":"Write a function to compute the following sum:\r\n\r\nSee also Cody Problem 46000.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 105px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 52.5px; transform-origin: 407px 52.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 143.008px 7.79167px; transform-origin: 143.008px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to compute the following sum:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 45px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22.5px; text-align: left; transform-origin: 384px 22.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y = sum(1/(n^2+an+b),{n,1,infinity})\" style=\"width: 122.5px; height: 45px;\" width=\"122.5\" height=\"45\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 29.175px 7.79167px; transform-origin: 29.175px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSee also \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46000-compute-the-harmonic-numbers\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration: underline; text-decoration-line: underline; \"\u003eCody Problem 46000\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.79167px; transform-origin: 1.94167px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = sumRecipQuad(a,b)\r\n y = sum(1/(n^2+a*n+b));\r\nend","test_suite":"%%\r\na = 3;\r\nb = 2;\r\ny_correct = 1/2;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 5;\r\nb = 6;\r\ny_correct = 1/3;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 7;\r\nb = 10;\r\ny_correct = 47/180;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 8;\r\nb = 15;\r\ny_correct = 9/40;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 10;\r\nb = 21;\r\ny_correct = 319/1680;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 9;\r\nb = 14;\r\ny_correct = 153/700;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 13;\r\nb = 22;\r\ny_correct = 42131/249480;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 14;\r\nb = 33;\r\ny_correct = 32891/221760;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 12;\r\nb = 35;\r\ny_correct = 13/84;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 16;\r\nb = 55;\r\ny_correct = 20417/166320;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 15;\r\nb = 26;\r\ny_correct = 605453/3963960;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 16;\r\nb = 39;\r\ny_correct = 485333/3603600;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 18;\r\nb = 65;\r\ny_correct = 323171/2882880;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 20;\r\nb = 91;\r\ny_correct = 30233/308880;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 24;\r\nb = 143;\r\ny_correct = 25/312;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 21;\r\nb = 38;\r\ny_correct = 158899519/1319157840;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 24;\r\nb = 95;\r\ny_correct = 19622959/217273056;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 30;\r\nb = 209;\r\ny_correct = 11171129/169303680;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 36;\r\nb = 323;\r\ny_correct = 37/684;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 25;\r\nb = 46;\r\ny_correct = 265842403/2498640144;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\nc = randi(50);\r\na = 2*c+1;\r\nb = c*(c+1);\r\ny_correct = 1/(c+1);\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\nfiletext = fileread('sumRecipQuad.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'assignin'); \r\nassert(~illegal)","published":true,"deleted":false,"likes_count":2,"comments_count":2,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":12,"test_suite_updated_at":"2021-07-03T14:09:11.000Z","rescore_all_solutions":true,"group_id":1,"created_at":"2021-06-27T14:48:37.000Z","updated_at":"2026-01-09T19:12:11.000Z","published_at":"2021-06-27T14:53:53.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to compute the following sum:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y = sum(1/(n^2+an+b),{n,1,infinity})\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey = \\\\sum_{n=1}^\\\\infty \\\\frac{1}{n^2+an+b}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSee also \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46000-compute-the-harmonic-numbers\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eCody Problem 46000\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":47874,"title":"Compute an integral of an exponential function","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 125px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 62.5px; transform-origin: 407px 62.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 72.3583px 7.91667px; transform-origin: 72.3583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThis problem builds on \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/47673-area-under-standard-normal-curve\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problem 47673\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 225.583px 7.91667px; transform-origin: 225.583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, in which Mehmet OZC asks us to compute the area under the standard normal curve. Write a function to compute the following integral:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 44px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22px; text-align: left; transform-origin: 384px 22px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 131px; height: 44px;\" width=\"131\" height=\"44\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 53.8083px 7.91667px; transform-origin: 53.8083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eYou may not use \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 30.8px 7.91667px; transform-origin: 30.8px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eintegral\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.1083px 7.91667px; transform-origin: 10.1083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e or \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.4px 7.91667px; transform-origin: 15.4px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003equad\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = IntegralExpFn(a,b,p)\r\n y = f(a,b,p);\r\nend","test_suite":"%%\r\na = log(2); b = 1; p = 1;\r\ny_correct = 1/2;\r\nassert(abs(IntegralExpFn(a,b,p)-y_correct) \u003c 1e-14)\r\n\r\n%%\r\na = Inf; b = 1; p = 1;\r\ny_correct = 1;\r\nassert(abs(IntegralExpFn(a,b,p)-y_correct) \u003c 1e-14)\r\n\r\n%%\r\na = Inf; b = 1; p = 2;\r\ny_correct = sqrt(pi)/2;\r\nassert(abs(IntegralExpFn(a,b,p)-y_correct) \u003c 1e-14)\r\n\r\n%%\r\na = 3; b = 1; p = 3;\r\ny_correct = 0.8929795115691813;\r\nassert(abs(IntegralExpFn(a,b,p)-y_correct) \u003c 1e-14)\r\n\r\n%%\r\na = 1; b = 1/2; p = 1.5;\r\ny_correct = 0.8278055502117507;\r\nassert(abs(IntegralExpFn(a,b,p)-y_correct) \u003c 1e-14)\r\n\r\n%%\r\na = Inf; b = randi(10); p = 1/2;\r\ny_correct = 2/b^2;\r\nassert(abs(IntegralExpFn(a,b,p)-y_correct) \u003c 1e-14)\r\n\r\n%%\r\na = 0; b = 4; p = 7/2;\r\ny_correct = 0;\r\nassert(abs(IntegralExpFn(a,b,p)-y_correct) \u003c 1e-14)\r\n\r\n%%\r\nfiletext = fileread('IntegralExpFn.m');\r\nillegalfns = ~isempty(strfind(filetext, 'integral')) || ~isempty(strfind(filetext, 'quad')); \r\nassert(~illegalfns,'Please do not use integral or quad')","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":6,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-12-10T04:36:31.000Z","updated_at":"2026-01-09T17:40:58.000Z","published_at":"2020-12-10T05:15:17.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis problem builds on \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/47673-area-under-standard-normal-curve\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem 47673\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, in which Mehmet OZC asks us to compute the area under the standard normal curve. Write a function to compute the following integral:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey = \\\\int_0^a \\\\exp(-b x^p) dx\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYou may not use \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eintegral\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr/\u003e\u003cw:t\u003e or \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003equad\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr/\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":46081,"title":"Set Soldner's constant","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 93px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 46.5px; transform-origin: 407px 46.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://mathworld.wolfram.com/SoldnersConstantDigits.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eSoldner's constant\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eμ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 199.375px 7.91667px; transform-origin: 199.375px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e= 1.451369234883381... is connected to the logarithmic integral \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"li(x)\" style=\"width: 30.5px; height: 19px;\" width=\"30.5\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 74.2833px 7.91667px; transform-origin: 74.2833px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, which is the subject of \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46066\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problem 46066\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 358.017px 7.91667px; transform-origin: 358.017px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSet Soldner's constant. The test suite puts up minor resistance against directly entering the number or using simple arithmetic. Of course, you can thwart the tests easily, but I encourage you to learn about the definition of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eμ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function mu = SoldnersConstant\r\n mu = ...;\r\nend","test_suite":"%%\r\nmu_correct = 1.451369234883381;\r\nassert(abs(SoldnersConstant-mu_correct) \u003c 1e-14)\r\n\r\n%%\r\nfiletext = fileread('SoldnersConstant.m');\r\nassert(isempty(strfind(filetext,'45136923488')), 'Please do not set the constant directly')\r\nbanfns = ~isempty(strfind(filetext, 'sum')) || ~isempty(strfind(filetext, 'plus')) ||...\r\n ~isempty(strfind(filetext, '+')) || ~isempty(strfind(filetext, 'minus')) || ...\r\n ~isempty(strfind(filetext, '-')) || ~isempty(strfind(filetext, 'diff')) || ...\r\n ~isempty(strfind(filetext, 'str2num'));\r\nassert(~banfns, 'Please do not set the constant indirectly with arithmetic')\r\n\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":7,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":16,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-07-30T02:45:56.000Z","updated_at":"2026-02-01T10:54:44.000Z","published_at":"2020-07-30T02:56:47.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://mathworld.wolfram.com/SoldnersConstantDigits.html\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eSoldner's constant\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"mu\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\mu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e= 1.451369234883381... is connected to the logarithmic integral \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"li(x)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e{\\\\rm li}(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, which is the subject of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46066\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem 46066\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSet Soldner's constant. The test suite puts up minor resistance against directly entering the number or using simple arithmetic. Of course, you can thwart the tests easily, but I encourage you to learn about the definition of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"mu\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\mu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":52120,"title":"Compute the fractional derivative","description":"Cody Problem 1370 asks us to compute the derivative of a polynomial. This problem extends that idea to fractional derivatives, which appear in some models of mixing in rivers and other applications. Denote the th derivative as . Then a familiar example from calculus would be . \r\nFractional calculus involves derivatives in which the order is not an integer. With and , then \r\n\r\nWrite a function that computes the fractional derivative of order of an expression of the form\r\n\r\nThe first input to the function will be a 2x matrix in which the first row is the coefficients and the second row is the exponents . The output should be in a similar form. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 256.55px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 128.275px; transform-origin: 407px 128.275px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/1370\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration: underline; text-decoration-line: underline; \"\u003eCody Problem 1370\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 294.442px 7.79167px; transform-origin: 294.442px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e asks us to compute the derivative of a polynomial. This problem extends that idea to fractional derivatives, which appear in some models of mixing in rivers and other applications. Denote the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eq\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 49.0083px 7.79167px; transform-origin: 49.0083px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eth derivative as \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"D^q x^a\" style=\"width: 32px; height: 19px;\" width=\"32\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.79167px; transform-origin: 3.88333px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Then a familiar example from calculus would be \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"D^2 x^3 = 6 x\" style=\"width: 65.5px; height: 19px;\" width=\"65.5\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.79167px; transform-origin: 3.88333px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 179.725px 7.79167px; transform-origin: 179.725px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFractional calculus involves derivatives in which the order \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eq\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 71.1667px 7.79167px; transform-origin: 71.1667px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is not an integer. With \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"q = 1/2\" style=\"width: 51.5px; height: 18.5px;\" width=\"51.5\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.79167px; transform-origin: 15.5583px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"a = 2\" style=\"width: 36.5px; height: 18px;\" width=\"36.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 19.4417px 7.79167px; transform-origin: 19.4417px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, then \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 36.9167px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.4583px; text-align: left; transform-origin: 384px 18.4583px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"D^{1/2} x^2 = 8 x^{3/2} / (3 sqrt(pi))\" style=\"width: 117px; height: 37px;\" width=\"117\" height=\"37\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 196.292px 7.79167px; transform-origin: 196.292px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that computes the fractional derivative of order \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eq\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 88.675px 7.79167px; transform-origin: 88.675px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e of an expression of the form\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 26px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 13px; text-align: left; transform-origin: 384px 13px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f(x) = c­1 x^a1 + c2 x^a2 + c3 x^a3 +...\" style=\"width: 204px; height: 26px;\" width=\"204\" height=\"26\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 43.6333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.8167px; text-align: left; transform-origin: 384px 21.8167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 124.85px 7.79167px; transform-origin: 124.85px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe first input to the function will be a 2x\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 143.775px 7.79167px; transform-origin: 143.775px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e matrix in which the first row is the coefficients \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABkAAAAoCAYAAAALz1FrAAAA9ElEQVRIie2VbQ2EMAyGHw9zMAMYOAUoOAc4wAEW0IAEPGABDWdh92NtRgjf2y65ZE/CDxi06du3BQqFQqEQjQVqoJXLpgz+AkZgBt5yPwAOaFIk6CTYsHquSWbAxCToJdC4cTbL2RSToJEgH7a1t3jZHldhJLjDy5UFrcIB1c1vDd559dmLI0Gqu3K8CYY4RKt40lSt5FSBmCSXmTieAYO3d5TrWkI16wZW+J6tJ90S5sqdJVCWze8kcY+vbk/vmu3tcEhFWIZXmqkrKMk+20N7mXQ7L7H8wJE6hNnWEARn6b8mi2TqxoGMjVcn3l2ohcI/8AWjzEUn2YFV5AAAAABJRU5ErkJggg==\" alt=\"ci\" style=\"width: 12.5px; height: 20px;\" width=\"12.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 83.625px 7.79167px; transform-origin: 83.625px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and the second row is the exponents \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABsAAAAoCAYAAAAPOoFWAAABF0lEQVRYhe2UUQ3DIBCGPw84mAEMoGAK5qAO6mAW0FAJeJiFaqiF7aF3KWNlkJQt2cKX8FD+9n7uehx0Op1O52ewwACMgJM9J8+2pUkAZjFzgAduwF3WqYXRRYLNgEm0e6R91Cg2ux41OkfB3I5+KujVGGCRQFPmHc16YT/raka2U+e6LIgeMrqlsks1q1wgFx1mzLzjRffvjGoCBcqZa2ZvSxyX8LyjD2yZN/1fqdmF9RpMPJfZ81oFS0WX2sgssJ7cSDC9b5qZlxV3rKPcPE/oyeM1sZVsSfZTrqINNWawlnCUD9K5Vxq+OjObDecchobzsoROlsPzsga9zHvXpjmzmBkqu/EI+r9ufKFBtIsPTZZO5894AM2JYzwwHTn5AAAAAElFTkSuQmCC\" alt=\"ai\" style=\"width: 13.5px; height: 20px;\" width=\"13.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 124.85px 7.79167px; transform-origin: 124.85px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The output should be in a similar form. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = fractDeriv(x,q)\r\n % x = 2xn matrix with coefficients in the first row and exponents in the second row\r\n % q = order of the derivative\r\n % y = output matrix with coefficients in the first row and exponents in the second row\r\n \r\n y = q*x^(q-1);\r\nend","test_suite":"%% Example from problem description\r\nx = [1; 2];\r\nq = 1/2;\r\ny_correct = [8/(3*sqrt(pi)); 3/2];\r\nassert(all(abs(fractDeriv(x,q)-y_correct)\u003c1e-10))\r\n\r\n%% Constant\r\nc = rand;\r\nx = [c; 0];\r\nq = 1/2;\r\ny_correct = [c/sqrt(pi); -1/2];\r\nassert(all(abs(fractDeriv(x,q)-y_correct)\u003c1e-10))\r\n\r\n%% Polynomial #1\r\nx = [3 -7 4; 2 1 0];\r\nq = 1/2;\r\ny_correct = [[8 -14 4]/sqrt(pi); 3/2 1/2 -1/2];\r\nassert(all(abs(fractDeriv(x,q)-y_correct)\u003c1e-10,'all'))\r\n\r\n%% Polynomial #2\r\nx = [1:4; 3:-1:0];\r\nq = 1/3;\r\ny_correct = [1.495438426033838 2.658557201837934 3.323196502297416 2.953952446486593; 8/3 5/3 2/3 -1/3];\r\nassert(all(abs(fractDeriv(x,q)-y_correct)\u003c1e-10,'all'))\r\n\r\n%% Quadratic term\r\nx = [7; 2];\r\nq = 3/2;\r\ny_correct = [28/sqrt(pi); 1/2];\r\nassert(all(abs(fractDeriv(x,q)-y_correct)\u003c1e-10))\r\n\r\n%% Two fractional derivatives amounting to a first derivative\r\nq = rand;\r\nx = [6; 5];\r\nyy_correct = [30; 4];\r\nassert(all(abs(fractDeriv(fractDeriv(x,q),1-q)-yy_correct)\u003c1e-10))\r\n\r\n%% Two fractional derivatives undoing each other\r\nq = rand;\r\nx = [5; 2];\r\nassert(all(abs(fractDeriv(fractDeriv(x,q),-q)-x)\u003c1e-10))\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":8,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-06-27T04:55:37.000Z","updated_at":"2026-01-09T18:33:26.000Z","published_at":"2021-06-27T05:00:18.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/1370\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eCody Problem 1370\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e asks us to compute the derivative of a polynomial. This problem extends that idea to fractional derivatives, which appear in some models of mixing in rivers and other applications. Denote the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"q\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eq\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eth derivative as \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"D^q x^a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eD^q x^a\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Then a familiar example from calculus would be \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"D^2 x^3 = 6 x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eD^2 x^3 = 6 x\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFractional calculus involves derivatives in which the order \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"q\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eq\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is not an integer. With \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"q = 1/2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eq = 1/2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a = 2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea = 2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, then \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"D^{1/2} x^2 = 8 x^{3/2} / (3 sqrt(pi))\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eD^{1/2} x^2 = \\\\frac{8}{3\\\\sqrt{\\\\pi}} x^{3/2}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that computes the fractional derivative of order \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"q\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eq\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e of an expression of the form\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(x) = c­1 x^a1 + c2 x^a2 + c3 x^a3 +...\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(x) = c­_1 x^{a_1} + c_2 x^{a_2} + c_3 x^{a_3} + \\\\dots \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe first input to the function will be a 2x\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e matrix in which the first row is the coefficients \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"ci\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec_i\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and the second row is the exponents \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"ai\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea_i\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The output should be in a similar form. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51745,"title":"Solve an ODE: equation A","description":"Write a function to solve the following ordinary differential equation: \r\n\r\nwith and . The function should return the values of at the specified values of . One application of this equation involves the propagation of internal gravity waves in a fluid with variable density gradient.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 118.6px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 59.3px; transform-origin: 407px 59.3px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 209.417px 8px; transform-origin: 209.417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve the following ordinary differential equation: \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 36.6px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.3px; text-align: left; transform-origin: 384px 18.3px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y\u0026quot;-xy = 0\" style=\"width: 74px; height: 36.5px;\" width=\"74\" height=\"36.5\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 43px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.5px; text-align: left; transform-origin: 384px 21.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14.3917px 8px; transform-origin: 14.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewith \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(x0) = y0\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(x1) = y1\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 128.742px 8px; transform-origin: 128.742px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The function should return the values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ey\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 80.9px 8px; transform-origin: 80.9px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at the specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 62.2333px 8px; transform-origin: 62.2333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. One application of this equation involves the propagation of internal gravity waves in a fluid with variable density gradient.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = solveODEA(x,x0,y0,x1,y1)\r\n y(x0) = y0; \r\n y(x1) = y1; \r\n y = f(x,x0,y0,x1,y1);\r\nend","test_suite":"%%\r\nx0 = -3; \r\ny0 = 1;\r\nx1 = 3;\r\ny1 = 0.2;\r\nx = linspace(x0,x1,7);\r\ny = solveODEA(x,x0,y0,x1,y1);\r\ny_correct = [1 -0.608544735138462 -1.416513846161609 -0.930561687980622 -0.339539877942166 -0.041385541117497 0.2];\r\nassert(all(abs(y-y_correct) \u003c 1e-13))\r\n\r\n%%\r\nx0 = -5; \r\ny0 = 2;\r\nx1 = 5;\r\ny1 = 0;\r\nx = linspace(x0,x1,11);\r\ny = solveODEA(x,x0,y0,x1,y1);\r\ny_correct = [2 -0.400646543833150 -2.159956361501116 1.296651996575315 3.053707895612597 2.024329503005815 0.771421025528832 0.199130370987114 0.037568752980204 0.005346964905914 0];\r\nassert(all(abs(y-y_correct) \u003c 1e-13))\r\n\r\n%%\r\nx0 = -4; \r\ny0 = -1;\r\nx1 = 2;\r\ny1 = 0.3;\r\nx = linspace(x0,x1,6);\r\ny = solveODEA(x,x0,y0,x1,y1);\r\ny_correct = [-1 -4.088820829832713 5.996215644909088 6.297137060461841 2.304927532867409 0.3];\r\nassert(all(abs(y-y_correct) \u003c 1e-13))\r\n\r\n%%\r\nx0 = -7; \r\ny0 = 0;\r\nx1 = 3;\r\ny1 = 0.1;\r\nx = linspace(x0,x1,6);\r\ny = solveODEA(x,x0,y0,x1,y1);\r\ny_correct = [0 -0.004972738967217 0.002891447704152 -0.005345046731767 0.007070410943182 0.1];\r\nassert(all(abs(y-y_correct) \u003c 1e-14))\r\n\r\n%% anti-cheating--product of two values\r\nx0 = -2; \r\ny0 = 1;\r\nx1 = 2;\r\ny1 = 0.1;\r\nx = [-1 1];\r\nz = prod(solveODEA(x,x0,y0,x1,y1)); \r\nz_correct = 1.336786968358133;\r\nassert(all(abs(z-z_correct) \u003c 1e-13))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":3,"created_by":46909,"edited_by":46909,"edited_at":"2022-06-15T13:16:15.000Z","deleted_by":null,"deleted_at":null,"solvers_count":7,"test_suite_updated_at":"2021-05-14T12:28:46.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-05-11T04:16:02.000Z","updated_at":"2025-09-02T13:25:13.000Z","published_at":"2021-05-11T04:19:47.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve the following ordinary differential equation: \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y\u0026quot;-xy = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\frac{d^2y}{dx^2} – x y = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewith \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(x0) = y0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(x_0) = y_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(x1) = y1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(x_1) = y_1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The function should return the values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at the specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. One application of this equation involves the propagation of internal gravity waves in a fluid with variable density gradient.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":58394,"title":"Integrate a product of gamma functions","description":"Write a function to compute the following integral:\r\n\r\nwhere and is the gamma function, the subject of Cody Problem 46025.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 104.1px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 52.05px; transform-origin: 407px 52.05px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 152.742px 8px; transform-origin: 152.742px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to compute the following integral:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 44px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22px; text-align: left; transform-origin: 384px 22px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 196.5px; height: 44px;\" width=\"196.5\" height=\"44\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21.1px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.55px; text-align: left; transform-origin: 384px 10.55px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 8px; transform-origin: 21.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 56px; height: 20px;\" width=\"56\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 29.5px; height: 18.5px;\" width=\"29.5\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 117.842px 8px; transform-origin: 117.842px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the gamma function, the subject of \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46025\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problem 46025\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = intGammaProduct(a)\r\n f = @(x) gamma(1+i*a*x)*gamma(1-i*a*x);\r\n y = trapz(x,f);\r\nend","test_suite":"%%\r\na = tan(1);\r\nI_correct = 1.008596722571773;\r\nassert(abs(intGammaProduct(a)-I_correct)\u003c1e-13)\r\n\r\n%%\r\na = sqrt(2);\r\nI_correct = 1.110720734539592;\r\nassert(abs(intGammaProduct(a)-I_correct)\u003c1e-13)\r\n\r\n%%\r\na = log(3);\r\nI_correct = 1.429800433690064;\r\nassert(abs(intGammaProduct(a)-I_correct)\u003c1e-13)\r\n\r\n%%\r\na = exp(4);\r\nI_correct = 0.028770138289325;\r\nassert(abs(intGammaProduct(a)-I_correct)\u003c1e-13)\r\n\r\n%%\r\na = sinh(5);\r\nI_correct = 0.021168845856719;\r\nassert(abs(intGammaProduct(a)-I_correct)\u003c1e-13)\r\n\r\n%%\r\na = asinh(6);\r\nI_correct = 0.630391294450658;\r\nassert(abs(intGammaProduct(a)-I_correct)\u003c1e-13)","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2023-06-03T19:50:53.000Z","deleted_by":null,"deleted_at":null,"solvers_count":8,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-06-03T14:18:04.000Z","updated_at":"2026-01-26T04:29:04.000Z","published_at":"2023-06-03T14:18:04.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to compute the following integral:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eI = \\\\int_{-\\\\infty}^\\\\infty \\\\Gamma(1+iax)\\\\Gamma(1-iax) dx\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ei = \\\\sqrt{-1}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Gamma(z)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the gamma function, the subject of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46025\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem 46025\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51137,"title":"Compute an integral of a product of sinusoids","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 104px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 52px; transform-origin: 407px 52px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 150.8px 7.91667px; transform-origin: 150.8px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to compute the following integral\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 44px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22px; text-align: left; transform-origin: 384px 22px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"I = integral(sin(x)^m cos(x)^n, {x,0,pi})\" style=\"width: 139px; height: 44px;\" width=\"139\" height=\"44\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 7.91667px; transform-origin: 21.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003em\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 96.5917px 7.91667px; transform-origin: 96.5917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are integers. You may not use \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 30.8px 7.91667px; transform-origin: 30.8px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eintegral\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.1083px 7.91667px; transform-origin: 10.1083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e or \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.4px 7.91667px; transform-origin: 15.4px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003equad\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 99.1917px 7.91667px; transform-origin: 99.1917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e but other functions are allowed.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = intSinmCosn(m,n)\r\n y = f(m,n);\r\nend","test_suite":"%%\r\nm = 1;\r\nn = 0;\r\ny_correct = 2;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 0;\r\nn = 1;\r\ny_correct = 0;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 0;\r\nn = 2;\r\ny_correct = pi/2;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 1;\r\nn = 2;\r\ny_correct = 2/3;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 2;\r\nn = 2;\r\ny_correct = pi/8;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 3;\r\nn = 2;\r\ny_correct = 4/15;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 0;\r\nn = 4;\r\ny_correct = 3*pi/8;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 1;\r\nn = 4;\r\ny_correct = 2/5;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 2;\r\nn = 4;\r\ny_correct = pi/16;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 3;\r\nn = 4;\r\ny_correct = 4/35;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 4;\r\nn = 4;\r\ny_correct = 3*pi/128;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 5;\r\nn = 4;\r\ny_correct = 16/315;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 6;\r\nn = 4;\r\ny_correct = 3*pi/256;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 7;\r\nn = 4;\r\ny_correct = 32/1155;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 0;\r\nn = 6;\r\ny_correct = 5*pi/16;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 1;\r\nn = 6;\r\ny_correct = 2/7;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 2;\r\nn = 6;\r\ny_correct = 5*pi/128;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 3;\r\nn = 6;\r\ny_correct = 4/63;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 4;\r\nn = 6;\r\ny_correct = 3*pi/256;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 5;\r\nn = 6;\r\ny_correct = 16/693;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 6;\r\nn = 6;\r\ny_correct = 5*pi/1024;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 7;\r\nn = 6;\r\ny_correct = 32/3003;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 0;\r\nn = 8;\r\ny_correct = 35*pi/128;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 1;\r\nn = 8;\r\ny_correct = 2/9;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 2;\r\nn = 8;\r\ny_correct = 7*pi/256;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 3;\r\nn = 8;\r\ny_correct = 4/99;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 4;\r\nn = 8;\r\ny_correct = 7*pi/1024;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 5;\r\nn = 8;\r\ny_correct = 16/1287;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 6;\r\nn = 8;\r\ny_correct = 5*pi/2048;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 7;\r\nn = 8;\r\ny_correct = 32/6435;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 2*randi(9);\r\nn = m+1;\r\ny_correct = 0;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 1;\r\nn = 22;\r\ny_correct = 2/23;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 1;\r\nn = 28;\r\ny_correct = 2/29;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nfiletext = fileread('intSinmCosn.m');\r\nillegalfns = ~isempty(strfind(filetext, 'integral')) || ~isempty(strfind(filetext, 'quad')); \r\nassert(~illegalfns,'Please do not use integral or quad')","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":10,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-03-23T00:29:43.000Z","updated_at":"2026-02-22T14:27:23.000Z","published_at":"2021-03-23T00:33:35.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to compute the following integral\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"I = integral(sin(x)^m cos(x)^n, {x,0,pi})\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eI = \\\\int_0^\\\\pi \\\\sin^mx\\\\cos^nxdx\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"m\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e are integers. You may not use \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eintegral\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e or \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003equad\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e but other functions are allowed.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":57452,"title":"Design a well field in an infinite aquifer","description":"A well field provides water for a community. The design of a well field involves a goal to meet a specified service demand (i.e., volume of water per time) with the constraint of lowering the water table by no more than , the maximum drawdown. Inputs to the design are properties of the aquifer (the hydraulic conductivity , the specific yield , and the initial saturated thickness ) and the radius of the well. \r\nThe Gupta/Chin method for designing a well field has the following steps:\r\nCompute , an initial estimate of the pumping rate, such that the drawdown at one well (i.e., at a distance ) is . Compute the transmissivity to be . Evaluate the drawdown at a time 1 year. Realize that for small values of the unconfined well function* can be approximated and compute the pumping rate from where and . \r\nCompute the number of wells by dividing the demand by the initial estimate of the pumping rate and rounding up to the nearest integer: \r\nSet the pumping rate to . \r\nArrange the wells so that they are equidistant from the central well.\r\nDetermine the distance between the central well and others so that the total drawdown at the central well is . In other words, add the drawdown from the central well to the drawdown from the other wells. If , then \r\n \r\n Write a function to design a well field using this method.\r\n\r\n\r\n*http://www.aqtesolv.com/neuman.htm","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 895.383px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 447.692px; transform-origin: 407px 447.692px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 87px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 43.5px; text-align: left; transform-origin: 384px 43.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 375.242px 8px; transform-origin: 375.242px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eA well field provides water for a community. The design of a well field involves a goal to meet a specified service demand \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"Q_d\" style=\"width: 19.5px; height: 20px;\" width=\"19.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 292.875px 8px; transform-origin: 292.875px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e (i.e., volume of water per time) with the constraint of lowering the water table by no more than \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"s_max\" style=\"width: 25.5px; height: 20px;\" width=\"25.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 47.8333px 8px; transform-origin: 47.8333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, the maximum drawdown. Inputs to the design are properties of the aquifer (the hydraulic conductivity \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eK\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 57.175px 8px; transform-origin: 57.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, the specific yield \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"S_y\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 29.1667px 8px; transform-origin: 29.1667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and the initial saturated thickness \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eb\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 50.5667px 8px; transform-origin: 50.5667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e) and the radius \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"r_w\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 37.3333px 8px; transform-origin: 37.3333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e of the well. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 226.008px 8px; transform-origin: 226.008px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe Gupta/Chin method for designing a well field has the following steps:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003col style=\"block-size: 231.783px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: decimal; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 391px 115.892px; transform-origin: 391px 115.892px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 104.05px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 52.025px; text-align: left; transform-origin: 363px 52.025px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 30.3417px 8px; transform-origin: 30.3417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eCompute \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"Q_w\" style=\"width: 20.5px; height: 20px;\" width=\"20.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 292.492px 8px; transform-origin: 292.492px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, an initial estimate of the pumping rate, such that the drawdown at one well (i.e., at a distance \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"r = r_w\" style=\"width: 40.5px; height: 20px;\" width=\"40.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 9.56667px 8px; transform-origin: 9.56667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e) is \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"s_max/2\" style=\"width: 40.5px; height: 20px;\" width=\"40.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 107.342px 8px; transform-origin: 107.342px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Compute the transmissivity to be \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"T = K(b-s_{max}/2)\" style=\"width: 116.5px; height: 20px;\" width=\"116.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 107.35px 8px; transform-origin: 107.35px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Evaluate the drawdown at a time \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"t = \" style=\"width: 22px; height: 18px;\" width=\"22\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 50.175px 8px; transform-origin: 50.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e 1 year. Realize that for small values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"u = S_y r^2/4Tt\" style=\"width: 81.5px; height: 21px;\" width=\"81.5\" height=\"21\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 243.133px 8px; transform-origin: 243.133px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e the unconfined well function* can be approximated and compute the pumping rate from \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"s_max/2 = (Q_w/4 pi T) W(u_w)\" style=\"width: 111.5px; height: 38px;\" width=\"111.5\" height=\"38\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 22.95px 8px; transform-origin: 22.95px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"W(u) = integral(exp(-x)/x,{x,0,infinity})\" style=\"width: 112px; height: 35.5px;\" width=\"112\" height=\"35.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-8px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"u_w = S_y r_w^2/4Tt\" style=\"width: 89.5px; height: 22px;\" width=\"89.5\" height=\"22\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 42.15px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 21.075px; text-align: left; transform-origin: 363px 21.075px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 356.708px 8px; transform-origin: 356.708px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eCompute the number of wells by dividing the demand by the initial estimate of the pumping rate and rounding up to the nearest integer: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"ceil(Q_d/Q_w)\" style=\"width: 88.5px; height: 20px;\" width=\"88.5\" height=\"20\"\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 21.7167px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.8583px; text-align: left; transform-origin: 363px 10.8583px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 74.675px 8px; transform-origin: 74.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSet the pumping rate to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"Q_0 = Q_d/N\" style=\"width: 73.5px; height: 20px;\" width=\"73.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 206.542px 8px; transform-origin: 206.542px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eArrange the wells so that they are equidistant from the central well.\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 43.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 21.7167px; text-align: left; transform-origin: 363px 21.7167px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 73.9083px 8px; transform-origin: 73.9083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eDetermine the distance \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eR\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 261px 8px; transform-origin: 261px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e between the central well and others so that the total drawdown at the central well is \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"s_max\" style=\"width: 25.5px; height: 20px;\" width=\"25.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. In other words, add the drawdown from the central well to the drawdown from the other wells. If \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg 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alt=\"u_R = S_y R^2/4Tt\" style=\"width: 91px; height: 21px;\" width=\"91\" height=\"21\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 19.4417px 8px; transform-origin: 19.4417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, then \u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e\u003cdiv style=\"block-size: 37.9px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.95px; text-align: left; transform-origin: 384px 18.95px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 23.3px 8px; transform-origin: 23.3px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg 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style=\"width: 227px; height: 38px;\" width=\"227\" height=\"38\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 174.133px 8px; transform-origin: 174.133px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e Write a function to design a well field using this 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\" data-image-state=\"image-loaded\" width=\"389\" height=\"374\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 118.5px 8px; transform-origin: 118.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e*http://www.aqtesolv.com/neuman.htm\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [N,Q0,R] = wellfield1(Qd,smax,rw,K,Sy,b)\r\n % Q0 = pumping rate (m3/d)\r\n % N = number of wells\r\n % R = distance between central well and other wells (m)\r\n N = randi(10);\r\n Q0 = round(Qd/N,'up');\r\n R = b;\r\nend","test_suite":"%%\r\nQd = 7000; % Demand (m3/d)\r\nsmax = 3; % Maximum drawdown (m)\r\nK = 50; % Hydraulic conductivity (m/d)\r\nSy = 0.2; % Specific yield \r\nb = 30; % Initial saturated thickness (m)\r\nrw = 0.3; % Radius of the well (m)\r\n[N,Q0,R] = wellfield1(Qd,smax,rw,K,Sy,b);\r\nQ0_correct = 1400;\r\nN_correct = 5;\r\nR_correct = 189.4;\r\nassert(abs(Q0-Q0_correct)\u003c1e-1)\r\nassert(isequal(N,N_correct))\r\nassert(abs(R-R_correct)\u003c1e-1)\r\n\r\n%%\r\nQd = 6000; % Demand (m3/d)\r\nsmax = 2.5; % Maximum drawdown (m)\r\nK = 60; % Hydraulic conductivity (m/d)\r\nSy = 0.22; % Specific yield \r\nb = 20; % Initial saturated thickness (m)\r\nrw = 0.25; % Radius of the well (m)\r\n[N,Q0,R] = wellfield1(Qd,smax,rw,K,Sy,b);\r\nQ0_correct = 857.1;\r\nN_correct = 7;\r\nR_correct = 297.6;\r\nassert(abs(Q0-Q0_correct)\u003c1e-1)\r\nassert(isequal(N,N_correct))\r\nassert(abs(R-R_correct)\u003c1e-1)\r\n\r\n%%\r\nQd = 10000; % Demand (m3/d)\r\nsmax = 4; % Maximum drawdown (m)\r\nK = 80; % Hydraulic conductivity (m/d)\r\nSy = 0.18; % Specific yield \r\nb = 25; % Initial saturated thickness (m)\r\nrw = 0.4; % Radius of the well (m)\r\n[N,Q0,R] = wellfield1(Qd,smax,rw,K,Sy,b);\r\nQ0_correct = 2500;\r\nN_correct = 4;\r\nR_correct = 117.6;\r\nassert(abs(Q0-Q0_correct)\u003c1e-1)\r\nassert(isequal(N,N_correct))\r\nassert(abs(R-R_correct)\u003c1e-1)\r\n\r\n%%\r\nQd = 12000; % Demand (m3/d)\r\nsmax = 3; % Maximum drawdown (m)\r\nK = 140; % Hydraulic conductivity (m/d)\r\nSy = 0.2; % Specific yield \r\nb = 25; % Initial saturated thickness (m)\r\nrw = 0.4; % Radius of the well (m)\r\n[N,Q0,R] = wellfield1(Qd,smax,rw,K,Sy,b);\r\nQ0_correct = 3000;\r\nN_correct = 4;\r\nR_correct = 78.2;\r\nassert(abs(Q0-Q0_correct)\u003c1e-1)","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2022-12-23T22:33:37.000Z","deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2022-12-23T00:54:17.000Z","updated_at":"2026-02-12T15:43:55.000Z","published_at":"2022-12-23T00:58:29.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA well field provides water for a community. The design of a well field involves a goal to meet a specified service demand \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Q_d\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eQ_d\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e (i.e., volume of water per time) with the constraint of lowering the water table by no more than \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"s_max\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es_{max}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, the maximum drawdown. Inputs to the design are properties of the aquifer (the hydraulic conductivity \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"K\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eK\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, the specific yield \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"S_y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eS_y\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and the initial saturated thickness \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"b\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eb\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e) and the radius \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"r_w\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003er_w\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e of the well. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe Gupta/Chin method for designing a well field has the following steps:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCompute \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Q_w\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eQ_w\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, an initial estimate of the pumping rate, such that the drawdown at one well (i.e., at a distance \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"r = r_w\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003er = r_w\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e) is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"s_max/2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es_{max}/2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Compute the transmissivity to be \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"T = K(b-s_{max}/2)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eT = K(b-s_{max}/2)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Evaluate the drawdown at a time \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"t = \\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003et =\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e 1 year. Realize that for small values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"u = S_y r^2/4Tt\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu = S_y r^2/4Tt\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e the unconfined well function* can be approximated and compute the pumping rate from \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"s_max/2 = (Q_w/4 pi T) W(u_w)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\frac{s_{max}}{2} = \\\\frac{Q_w}{4\\\\pi T} W(u_w)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"W(u) = integral(exp(-x)/x,{x,0,infinity})\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eW(u) = \\\\int_u^\\\\infty \\\\frac{e^{-x}}{x} dx\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"u_w = S_y r_w^2/4Tt\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu_w = S_y r_w^2/4 T t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCompute the number of wells by dividing the demand by the initial estimate of the pumping rate and rounding up to the nearest integer: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"ceil(Q_d/Q_w)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eN = \\\\lceil Q_d/Q_w\\\\rceil\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSet the pumping rate to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Q_0 = Q_d/N\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eQ_0 = Q_d/N\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eArrange the wells so that they are equidistant from the central well.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eDetermine the distance \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"R\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eR\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e between the central well and others so that the total drawdown at the central well is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"s_max\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es_{max}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. In other words, add the drawdown from the central well to the drawdown from the other wells. If \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"u_R = S_y R^2/4Tt\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu_R = S_y R^2/4Tt\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, then \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es_{max} = \\\\frac{Q_0}{4\\\\pi T}\\\\left[W(u_w) + (N-1) W(u_R)\\\\right]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e Write a function to design a well field using this method.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"374\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"389\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc 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an ODE: equation C","description":"Write a function to solve the following ordinary differential equation: \r\n\r\nwith and . The parameter is a constant. The function should return the values of at the specified values of .","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 118.583px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 59.2917px; transform-origin: 407px 59.2917px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 209.417px 7.91667px; transform-origin: 209.417px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve the following ordinary differential equation: \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 37.3333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.6667px; text-align: left; transform-origin: 384px 18.6667px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"(1-x^2) y\u0026quot; - x y' + a^2 y = 0\" style=\"width: 179.5px; height: 37.5px;\" width=\"179.5\" height=\"37.5\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.25px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.125px; text-align: left; transform-origin: 384px 21.125px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14.3917px 7.91667px; transform-origin: 14.3917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewith \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(x0) = y0\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(x1) = y1\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 51.725px 7.91667px; transform-origin: 51.725px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The parameter \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 169.967px 7.91667px; transform-origin: 169.967px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a constant. The function should return the values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ey\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 50.95px 7.91667px; transform-origin: 50.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at the specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = solveODEC(x,a,Xbc,bc)\r\n% y = values of the function\r\n% x = independent variable\r\n% a = parameter in the equation\r\n% Xbc = [x0 x1], values of x where the boundary conditions are specified\r\n% bc = [y0 y1], values of the function at x0 and x1, respectively\r\n\r\n y = f(x,a,Xbc,bc);\r\nend","test_suite":"%% \r\nx = [-1/3 -1/4 0 1/4 1/3];\r\na = 1;\r\nXbc = [-1/2 1/2];\r\nbc = [2 4];\r\ny = solveODEC(x,a,Xbc,bc);\r\ny_correct = [2.599319657044238 2.854101966249684 3.464101615137754 3.854101966249684 3.932652990377571];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%% \r\nx = [-0.6 -0.2 -0.05 0.12 0.2];\r\na = sqrt(3);\r\nXbc = [-0.7 0.3];\r\nbc = [0 1];\r\ny = solveODEC(x,a,Xbc,bc);\r\ny_correct = [0.237055225759061 0.877546669526703 0.995431932094838 1.046546024897035 1.039090864471318];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%% \r\nx = [-0.8 -0.6 -0.3 -0.15 0.1 0.25 0.375];\r\na = 2;\r\nXbc = [-0.8 0.4];\r\nbc = [-1 3];\r\ny = solveODEC(x,a,Xbc,bc);\r\ny_correct = [-1 1.68647951290722 4.138417234449975 4.687455882945745 4.630189304757032 4.024530246664777 3.199461161409147];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%% \r\nx = [-0.7 -0.45 -0.2 0.05 0.3 0.55 0.8];\r\na = pi;\r\nXbc = [-0.9 0.9];\r\nbc = [-1 1];\r\ny = solveODEC(x,a,Xbc,bc);\r\ny_correct = [1.764854605944604 2.706629373469937 1.6090059417224112 -0.4259046519843118 -2.225041323571773 -2.630850806938173 -0.6162122684969365];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%%\r\na = 5;\r\nXbc = [-1/2 1/2]; \r\nbc = [-1 1];\r\ny = 1;\r\nx = fzero(@(z) solveODEC(z,a,Xbc,bc)-y,0);\r\nx_correct = 0.104528463267653;\r\nassert(abs(x-x_correct)\u003c1e-13)\r\n\r\n%%\r\na = 5;\r\nXbc = [-1/2 1/2]; \r\nbc = [-1 1];\r\ny = 1;\r\nx = fzero(@(z) solveODEC(z,a,Xbc,bc)-y,0);\r\nx_correct = 0.104528463267653;\r\nassert(abs(x-x_correct)\u003c1e-13)\r\n\r\n%%\r\na = 2*randi(4)+1;\r\nXbc = [-3/4 3/4]; \r\nbc = [-2 2];\r\nx = rand-3/4;\r\nym = solveODEC(-x,a,Xbc,bc);\r\nyp = solveODEC(x,a,Xbc,bc);\r\nassert(abs(ym+yp)\u003c1e-13)","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-05-25T04:18:45.000Z","updated_at":"2021-05-25T04:23:56.000Z","published_at":"2021-05-25T04:23:56.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve the following ordinary differential equation: \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"(1-x^2) y\u0026quot; - x y' + a^2 y = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e(1-x^2)\\\\frac{d^2y}{dx^2} -x \\\\frac{dy}{dx}+a^2 y = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewith \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(x0) = y0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(x_0) = y_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(x1) = y1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(x_1) = y_1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The parameter \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a constant. The function should return the values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at the specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60748,"title":"Extend the digamma function to negative arguments","description":"While solving one of Ramon Villamangca’s problems, I needed the value of , where is the digamma function. However, MATLAB’s function psi does not work for negative arguments. \r\nWrite a function that extends the digamma function to negative arguments. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 72px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 36px; transform-origin: 407px 36px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWhile solving one of Ramon Villamangca’s problems, I needed the value of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"57.5\" height=\"18\" style=\"width: 57.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"32\" height=\"18\" style=\"width: 32px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the digamma function. However, MATLAB’s function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003epsi\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e does not work for negative arguments. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that extends the digamma function to negative arguments. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = newpsi(x)\r\n y = -psi(abs(x));\r\nend","test_suite":"%%\r\nx = -1/2;\r\ny = newpsi(x);\r\ny_correct = 0.0364899739785765;\r\nassert(abs(y-y_correct)\u003c1e-13)\r\n\r\n%%\r\nx = -1/3;\r\ny = newpsi(x);\r\ny_correct = 1.68176558421341;\r\nassert(abs(y-y_correct)\u003c1e-13)\r\n\r\n%%\r\nx = -1/4;\r\ny = newpsi(x);\r\ny_correct = 2.91413912021353;\r\nassert(abs(y-y_correct)\u003c1e-11)\r\n\r\n%%\r\nx = -1/5;\r\ny = newpsi(x);\r\ny_correct = 4.03499143329386;\r\nassert(abs(y-y_correct)\u003c1e-11)\r\n\r\n%%\r\nx = -2/7;\r\ny = newpsi(x);\r\ny_correct = 2.31981855966050;\r\nassert(abs(y-y_correct)\u003c1e-11)\r\n\r\n%%\r\nx = -3/8;\r\ny = newpsi(x);\r\ny_correct = 1.21395790209010;\r\nassert(abs(y-y_correct)\u003c1e-11)\r\n\r\n%%\r\nx = -4/9;\r\ny = newpsi(x);\r\ny_correct = 0.537183359660484;\r\nassert(abs(y-y_correct)\u003c1e-11)\r\n\r\n%%\r\nx = -5/11;\r\ny = newpsi(x);\r\ny_correct = 0.444812910020638;\r\nassert(abs(y-y_correct)\u003c1e-11)\r\n\r\n%%\r\nx = -17/13;\r\ny = newpsi(x);\r\ny_correct = 2.77268958155960;\r\nassert(abs(y-y_correct)\u003c1e-11)\r\n\r\n%%\r\nx = -37/19;\r\ny = newpsi(x);\r\ny_correct = -17.9247506862983;\r\nassert(abs(y-y_correct)\u003c1e-11)\r\n\r\n%%\r\nx = -exp(1);\r\ny = newpsi(x);\r\ny_correct = -1.39770931071902;\r\nassert(abs(y-y_correct)\u003c1e-11)\r\n\r\n%%\r\nx = -pi;\r\ny = newpsi(x);\r\ny_correct = 7.88595238538549;\r\nassert(abs(y-y_correct)\u003c1e-11)\r\n\r\n%%\r\nx = -sqrt(73);\r\ny = newpsi(x);\r\ny_correct = 1.76552281819428;\r\nassert(abs(y-y_correct)\u003c1e-11)\r\n\r\n%%\r\nx = -sqrt(273);\r\ny = newpsi(x);\r\ny_correct = 2.61015605350263;\r\nassert(abs(y-y_correct)\u003c1e-11)\r\n\r\n%%\r\nx = -sqrt(7073);\r\ny = newpsi(x);\r\ny_correct = 13.9912915572908;\r\nassert(abs(y-y_correct)\u003c1e-11)\r\n\r\n%%\r\nx = 1;\r\ny = newpsi(x);\r\ny_correct = -0.577215664901532;\r\nassert(abs(y-y_correct)\u003c1e-11)\r\n\r\n%%\r\nx = sqrt(7073);\r\ny = newpsi(x);\r\ny_correct = 4.426062993434546;\r\nassert(abs(y-y_correct)\u003c1e-11)\r\n\r\n%%\r\nx = -[293/37 263/47 233/67 223/97 193/107 173/127 163/137];\r\ny = arrayfun(@newpsi,x);\r\ns = sum(y);\r\np = prod(y);\r\ns_correct = -0.500409630494963;\r\np_correct = 1714.480727245714;\r\nassert(abs(s-s_correct)\u003c1e-11)\r\nassert(abs(p-p_correct)\u003c1e-8)","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2024-10-13T04:37:44.000Z","deleted_by":null,"deleted_at":null,"solvers_count":6,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2024-10-13T04:28:19.000Z","updated_at":"2025-10-02T13:47:28.000Z","published_at":"2024-10-13T04:37:44.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWhile solving one of Ramon Villamangca’s problems, I needed the value of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\psi(-1/2)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\psi(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the digamma function. However, MATLAB’s function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003epsi\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e does not work for negative arguments. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that extends the digamma function to negative arguments. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":46117,"title":"Test approximations of the prime counting function","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 162.817px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 81.4083px; transform-origin: 407px 81.4083px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 90px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 45px; text-align: left; transform-origin: 384px 45px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/241\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problem 241\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 61.4583px 7.79167px; transform-origin: 61.4583px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, which is based on \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"http://projecteuler.net/problem=7\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eProject Euler Problem 7\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 73.1083px 7.79167px; transform-origin: 73.1083px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, asks us to identify the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eN\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 91px 7.79167px; transform-origin: 91px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eth prime number. That is, the problem seeks the inverse of the prime counting function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"pi(n)\" style=\"width: 31px; height: 18.5px;\" width=\"31\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 185.15px 7.79167px; transform-origin: 185.15px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, which provides the number of primes less than or equal to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.8833px 7.79167px; transform-origin: 17.8833px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://mathworld.wolfram.com/PrimeNumberTheorem.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime Number Theorem\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 87.9px 7.79167px; transform-origin: 87.9px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e gives approximate forms of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"pi(n)\" style=\"width: 31px; height: 18.5px;\" width=\"31\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 29.5583px 7.79167px; transform-origin: 29.5583px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e for large \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 96.8583px 7.79167px; transform-origin: 96.8583px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Two such approximations are \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"n/ln(n)\" style=\"width: 52px; height: 18.5px;\" width=\"52\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 27.225px 7.79167px; transform-origin: 27.225px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and the \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Logarithmic_integral_function#Offset_logarithmic_integral\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"perspective-origin: 77.675px 7.79167px; transform-origin: 77.675px 7.79167px; \"\u003eoffset logarithmic integral\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.79167px; transform-origin: 1.94167px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"Li(n) = li(n) - li(2)\" style=\"width: 126px; height: 18.5px;\" width=\"126\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24.8917px 7.79167px; transform-origin: 24.8917px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-8px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"li(x) = integral of 1/ln(t) from t = 0 to t = infinity\" style=\"width: 122px; height: 27px;\" width=\"122\" height=\"27\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 18.6667px 7.79167px; transform-origin: 18.6667px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e (See \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46066-evaluate-the-logarithmic-integral\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problem 46066\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4.275px 7.79167px; transform-origin: 4.275px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e).\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63.8167px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.9083px; text-align: left; transform-origin: 384px 31.9083px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 160.642px 7.79167px; transform-origin: 160.642px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eTest these approximations by computing two ratios: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"r1 = [n/ln(n)]/pi(n)\" style=\"width: 128px; height: 20px;\" width=\"128\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.79167px; transform-origin: 15.5583px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"r2 = Li(n)/pi(n)\" style=\"width: 100.5px; height: 20px;\" width=\"100.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 57.95px 7.79167px; transform-origin: 57.95px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Do not round the approximations to integers. For \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"n = 100\" style=\"width: 51.5px; height: 18px;\" width=\"51.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 257.5px 7.79167px; transform-origin: 257.5px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, you will find that the first approximation is about 13% low and the second is about 16% high. However, for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"n = 10^8\" style=\"width: 49px; height: 19px;\" width=\"49\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 216.658px 7.79167px; transform-origin: 216.658px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, the first approximation is 6% low and the second is only 0.01% high. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [r1,r2] = primeCount(n)\r\n r1 = (n/ln(n))/primepi(n);\r\n r2 = Li(n)/primepi(n);\r\nend","test_suite":"%%\r\nn = 1e2;\r\nr1_correct = 0.86859;\r\nr2_correct = 1.16324;\r\n[r1,r2] = primeCount(n);\r\nassert(isequal(round(r1,5),r1_correct) \u0026\u0026 isequal(round(r2,5),r2_correct))\r\n\r\n%%\r\nn = 1e4;\r\nr1_correct = 0.88343;\r\nr2_correct = 1.01309;\r\n[r1,r2] = primeCount(n);\r\nassert(isequal(round(r1,5),r1_correct) \u0026\u0026 isequal(round(r2,5),r2_correct))\r\n\r\n%%\r\nn = 1e6;\r\nr1_correct = 0.92209;\r\nr2_correct = 1.00164;\r\n[r1,r2] = primeCount(n);\r\nassert(isequal(round(r1,5),r1_correct) \u0026\u0026 isequal(round(r2,5),r2_correct))\r\n\r\n%%\r\nn = 1e8;\r\nr1_correct = 0.94224;\r\nr2_correct = 1.00013;\r\n[r1,r2] = primeCount(n);\r\nassert(isequal(round(r1,5),r1_correct) \u0026\u0026 isequal(round(r2,5),r2_correct))\r\n\r\n%%\r\nn = 1e5;\r\n[r1,r2] = primeCount(n);\r\ns1 = floor(1e5*round(r1,5));\r\ns2 = floor(1e5*round(r2,5));\r\nbxo_correct = 59814;\r\nassert(isequal(bitxor(s1,s2),bxo_correct))","published":true,"deleted":false,"likes_count":1,"comments_count":3,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":16,"test_suite_updated_at":"2021-01-03T15:27:20.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2020-08-07T15:57:52.000Z","updated_at":"2025-08-18T01:32:16.000Z","published_at":"2020-08-07T16:33:15.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/241\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem 241\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, which is based on \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://projecteuler.net/problem=7\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eProject Euler Problem 7\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, asks us to identify the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"N\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eN\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eth prime number. That is, the problem seeks the inverse of the prime counting function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"pi(n)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi(n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, which provides the number of primes less than or equal to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://mathworld.wolfram.com/PrimeNumberTheorem.html\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime Number Theorem\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e gives approximate forms of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"pi(n)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi(n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for large \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Two such approximations are \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n/ln(n)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en/\\\\ln(n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Logarithmic_integral_function#Offset_logarithmic_integral\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eoffset logarithmic integral\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Li(n) = li(n) - li(2)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e{\\\\rm Li(n)} ={\\\\rm li(n)} - {\\\\rm li(2)}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"li(x) = integral of 1/ln(t) from t = 0 to t = infinity\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e{\\\\rm li}(x) = \\\\int_0^\\\\infty dt/\\\\ln(t)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e (See \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46066-evaluate-the-logarithmic-integral\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem 46066\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTest these approximations by computing two ratios: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"r1 = [n/ln(n)]/pi(n)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003er_1 =[n/\\\\ln(n)]/ \\\\pi(n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"r2 = Li(n)/pi(n)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003er_2 = {\\\\rm Li}(n)/\\\\pi(n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Do not round the approximations to integers. For \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n = 100\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en = 100\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, you will find that the first approximation is about 13% low and the second is about 16% high. However, for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n = 10^8\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en = 10^8\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, the first approximation is 6% low and the second is only 0.01% high. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":59571,"title":"Compute a sum involving the zeta function","description":"Write a function to compute the sum\r\n\r\nfor , where is the zeta function, the subject of Cody Problems 45939, 45988, and 45997.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 105px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 52.5px; transform-origin: 407px 52.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 111.883px 8px; transform-origin: 111.883px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to compute the sum\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 45px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22.5px; text-align: left; transform-origin: 384px 22.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"137.5\" height=\"45\" alt=\"S(x) = sum(zeta(n+1) x^n) for n = 1 to n = inf\" style=\"width: 137.5px; height: 45px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.1083px 8px; transform-origin: 10.1083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003efor \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"46.5\" height=\"18.5\" alt=\"|x| \u003c 1\" style=\"width: 46.5px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24.8917px 8px; transform-origin: 24.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEQAAAAlCAYAAAD7u09NAAAEdklEQVRoQ+2ZS6hPURTG3TkRIwYUBpRXeZZMDCgpSXlEuiOPgQxEMTAwoOgOZOCRgYwwkZQiMRB5TJBigDJh5FHM+X63s7Tvtl/nnHsu6pz6uqf/3mudvb/13PsOjOufEQwM9HyMZKAnxPOInpB/gJClWsNM4WrH4TpL+pfU/c5YewhknBHWCV86JgT1J4UZwpbSb9UhZI+UbhTWlir35kHGZWHlGJFhnzdPLCKlhJDJ0vxQeCkcEd5VX4Kg/cIcYa9wLkEUOj4Im4Q7DQltI/ZYwpcyaxzWX0IIDG8WZldkmKUhgueHsFp4llgxC4KQIiu12XlEljXfExZVe4h+ooSQ25JeI+AFPKeE8dX7G/3FS1JWRxYdRmgH+y1SiWHJJytSs0sIgd0bwtRKER7xSLguXBNyyfFve4ft3wyzTD9EvbmEEBQaKcf0nsoVPvnIPRW2Cl2X2RI3+ahJD4Ro6JYQYmQMZkIjtKDz+nGXMKXAk0o21HYORqHkT4gpyhFCdXgl0DucaLCa15XM3AayXYgcltLjQjRscoSQDL+lXCyz6p8afyKkEhmk01EuFqgCJL6JgvUrxP5pgapG/nIrhTV6y/X7J2FexhMtj0TbhBQh9BlHhVWC9R51rGb5g6p0KCFohBzQHBbMc0HYLdgabuqd0PPH0H1LoC1wx2KfM0Kia4oRYo0U7tUkVFhQ9uPeqtn82eo3umHOO5T09QIGwdt4aA6/VmM79ZeKYWM58pFnLtUxmFhjhHAG2CHkXDBmiSaEWAImLA56ZBi56MXdXaLcMYjMdcLJMI4R8l2K23hHE0JIwOQJcs50wQ1VS4bkCZ5BZ+MYDwJLcoh5SDSvhQjBla4IyQYm5RrVWJ2QIUQ/V3LmIW6/Y90yUyy/2BJo/EiqeEbu4Gnf8XX83k6IEGr1AqHtqZT7iLdCSVy7+YONbRPcDhiP5bgQ8gI3t+TyXdZIIUJgnLI3Gr1DSdnFOnaA5N33TDdHkFDdTbtjJR5t86Odc4gQNoElphWERW4K5JIPcrrMA4ht//LIcgShRI/ieo6Nccgs8WibHz1oxghhoyUZO0eIJcNU657yAPRbsg3lCMsfVkYJvfdCrNKQi+anDBQihAOQnWxR/EKg7t8VUnceIXIsj6QukKzcIu9bzuQZC+mw/MEYRHD4TF1PMj+Z00KEmFvFrB9KeilPwYo8sfbdLbf+HDfZ+mRZJ4xuQo1nuxDrqk1X8l4m15iZp/gbzp1P3PmpCyLXA/yEiQ4rt6EcgexzgepDyLjXmyEDYRi8nSNB9Mkd7hDEEjyEi3tZVOdIT1gsFJK3VamFthyjt7oo+En5D7UlhPhCTdp6u6jm1FrngqklD8PitS64mxBCzzBJyHWF/mbsbmVD5W2jsdkSHcU37ijLEWIVwFpdyiixmrtljy0UUjiu7xsjUjDe/TpemSPEypq7wVDyK7GUzbH7j9yptI7O0FySLt+q1SrkCKFU2b8d2MCQ0PVG2hLRSj5HSCvl/6NwT4hntZ6QnpB0IP8C3IMRNWT+n34AAAAASUVORK5CYII=\" width=\"34\" height=\"18.5\" alt=\"zeta(m)\" style=\"width: 34px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 157.517px 8px; transform-origin: 157.517px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the zeta function, the subject of Cody Problems \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/45939\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003e45939\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/45988\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003e45988\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.5px 8px; transform-origin: 17.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/45997\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003e45997\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function S = zetasum(x)\r\n n = 1:Inf;\r\n S = sum(zeta(n+1).*x.^n);\r\nend","test_suite":"%%\r\nx = 1/2;\r\nS = zetasum(x);\r\nS_correct = 1.386294361119891;\r\nassert(abs(S-S_correct)/S_correct\u003c1e-12)\r\n\r\n%%\r\nx = 2/3;\r\nS = zetasum(x);\r\nS_correct = 2.554818115119273;\r\nassert(abs(S-S_correct)/S_correct\u003c1e-12)\r\n\r\n%%\r\nx = 3/4;\r\nS = zetasum(x);\r\nS_correct = 3.650237868474732;\r\nassert(abs(S-S_correct)/S_correct\u003c1e-12)\r\n\r\n%%\r\nx = 5/6;\r\nS = zetasum(x);\r\nS_correct = 5.754911840473381;\r\nassert(abs(S-S_correct)/S_correct\u003c1e-12)\r\n\r\n%%\r\nx = 7/8;\r\nS = zetasum(x);\r\nS_correct = 7.811276998394322;\r\nassert(abs(S-S_correct)/S_correct\u003c1e-12)\r\n\r\n%%\r\nx = 8/9;\r\nS = zetasum(x);\r\nS_correct = 8.83072761223029;\r\nassert(abs(S-S_correct)/S_correct\u003c1e-12)\r\n\r\n%%\r\nx = 9/10;\r\nS = zetasum(x);\r\nS_correct = 9.84653927550954;\r\nassert(abs(S-S_correct)/S_correct\u003c1e-12)\r\n\r\n%%\r\nx = 10/11;\r\nS = zetasum(x);\r\nS_correct = 10.85964675709217;\r\nassert(abs(S-S_correct)/S_correct\u003c1e-12)\r\n\r\n%%\r\nx = 11/12;\r\nS = zetasum(x);\r\nS_correct = 11.87068966352595;\r\nassert(abs(S-S_correct)/S_correct\u003c1e-12)\r\n\r\n%% \r\nx = 0.232931374143;\r\nSS = zetasum(zetasum(x));\r\nSS_correct = 1.227707484938568;\r\nassert(abs(SS-SS_correct)/SS_correct\u003c1e-12)\r\n\r\n%%\r\nx = 1./primes(20);\r\ny = sum(arrayfun(@zetasum,x));\r\ny_correct = 3.2640541637441439;\r\nassert(abs(y-y_correct)/y_correct\u003c1e-12)\r\n\r\n%%\r\nfiletext = fileread('zetasum.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert') || contains(filetext,'regexp') || contains(filetext,'switch'); \r\nassert(~illegal)","published":true,"deleted":false,"likes_count":1,"comments_count":2,"created_by":46909,"edited_by":46909,"edited_at":"2024-01-20T17:57:17.000Z","deleted_by":null,"deleted_at":null,"solvers_count":7,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2024-01-20T13:37:10.000Z","updated_at":"2026-03-04T13:56:18.000Z","published_at":"2024-01-20T13:40:38.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to compute the sum\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"S(x) = sum(zeta(n+1) x^n) for n = 1 to n = inf\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eS(x) = \\\\sum_{n=1}^\\\\infty \\\\zeta(n+1) x^n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003efor \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"|x| \u0026lt; 1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e|x| \u0026lt; 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"zeta(m)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\zeta(m)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the zeta function, the subject of Cody Problems \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/45939\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e45939\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/45988\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e45988\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/45997\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e45997\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":59981,"title":"Compute the Ramanujan tau function","description":"The Ramanujan tau function is defined by the relation\r\n\r\nwhere . The first few values of are 1, -24, 252, -1472, and 4830. \r\nWrite a function to compute the Ramanujan tau function.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 125px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 62.5px; transform-origin: 407px 62.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 89.8583px 8px; transform-origin: 89.8583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe Ramanujan tau function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"28.5\" height=\"18.5\" alt=\"tau(n)\" style=\"width: 28.5px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 77.8px 8px; transform-origin: 77.8px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is defined by the relation\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 35px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 17.5px; text-align: left; transform-origin: 384px 17.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"172\" height=\"35\" alt=\"Sum[tau(n)q^n,{n,1,inf}] = q Product[(1-q^n)^24,{n,1,inf}]\" style=\"width: 172px; height: 35px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 8px; transform-origin: 21.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"88\" height=\"18.5\" alt=\"q = exp(2 pi i z)\" style=\"width: 88px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 73.8833px 8px; transform-origin: 73.8833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The first few values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"28.5\" height=\"18.5\" alt=\"tau(n)\" style=\"width: 28.5px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 106.183px 8px; transform-origin: 106.183px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are 1, -24, 252, -1472, and 4830. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 98.6583px 8px; transform-origin: 98.6583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to compute the \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://mathworld.wolfram.com/TauFunction.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003eRamanujan tau function\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = Ramanujantau(n)\r\n y = q*prod((1-q^n)^24)/sum(q^n);\r\nend","test_suite":"%%\r\nassert(isequal(Ramanujantau(1),1))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(2),-24))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(7),-16744))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(14),401856))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(22),-12830688))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(51),-1740295368))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(86),411016992))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(147),-427635232164))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(243),13400796651732))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(260),4107578522880))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(300),9458784518400))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(325),14731871253050))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(400),-25171202969600))\r\n\r\n%%\r\nassert(isequal(sum(arrayfun(@Ramanujantau,1:300)),-33462718906943))\r\n\r\n%%\r\nfiletext = fileread('Ramanujantau.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert') || contains(filetext, 'read'); \r\nassert(~illegal)","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2024-04-26T13:15:52.000Z","deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2024-04-25T03:28:46.000Z","updated_at":"2024-04-26T13:15:52.000Z","published_at":"2024-04-25T03:28:58.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe Ramanujan tau function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"tau(n)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\tau(n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is defined by the relation\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Sum[tau(n)q^n,{n,1,inf}] = q Product[(1-q^n)^24,{n,1,inf}]\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\sum_{n\\\\ge1} \\\\tau(n) q^n = q\\\\prod_{n\\\\ge1}(1-q^n)^{24}\\n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"q = exp(2 pi i z)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eq = \\\\exp(2\\\\pi i z)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The first few values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"tau(n)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\tau(n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e are 1, -24, 252, -1472, and 4830. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to compute the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://mathworld.wolfram.com/TauFunction.html\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eRamanujan tau function\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60633,"title":"Sum the reciprocals of polygonal numbers","description":"As explained in Cody Problem 60571, a polygonal number is the number of dots arranged in the shape of a regular polygon. For example, 15 is a triangular number because dots can be arranged in the shape of a triangle with rows of 1, 2, 3, 4, and 5 dots. The number 16 is a square number because dots can be arranged in four rows of four. \r\nWrite a function to sum the reciprocals of polygonal numbers. In particular, compute\r\n\r\nwhere is the th -gonal number (i.e., the th number corresponding to a regular polygon with sides). ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 178px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 89px; transform-origin: 407px 89px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 49.4083px 8px; transform-origin: 49.4083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAs explained in \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/60571\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003eCody Problem 60571\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 9.71667px 8px; transform-origin: 9.71667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, a \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Polygonal_number\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003epolygonal number\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 203.483px 8px; transform-origin: 203.483px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the number of dots arranged in the shape of a regular polygon. For example, 15 is a triangular number because dots can be arranged in the shape of a triangle with rows of 1, 2, 3, 4, and 5 dots. The number 16 is a square number because dots can be arranged in four rows of four. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 258.533px 8px; transform-origin: 258.533px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to sum the reciprocals of polygonal numbers. In particular, compute\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 45px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22.5px; text-align: left; transform-origin: 384px 22.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"74\" height=\"45\" alt=\"y = sum[1/P_n,s,{n,1,inf}]\" style=\"width: 74px; height: 45px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 11px; text-align: left; transform-origin: 384px 11px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 8px; transform-origin: 21.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"23.5\" height=\"20\" alt=\"P_n,s\" style=\"width: 23.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 20.6083px 8px; transform-origin: 20.6083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.775px 8px; transform-origin: 7.775px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eth \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003es\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 74.2917px 8px; transform-origin: 74.2917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e-gonal number (i.e., the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 157.942px 8px; transform-origin: 157.942px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eth number corresponding to a regular polygon with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003es\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24.5px 8px; transform-origin: 24.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e sides). \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = sumRecipPolyNum(s)\r\n y = sum(1/s);\r\n","test_suite":"%%\r\nn = 3;\r\ny = sumRecipPolyNum(n);\r\ny_correct = 2;\r\nassert(abs(y-y_correct)\u003c1e-13)\r\n\r\n%%\r\nn = 4;\r\ny = sumRecipPolyNum(n);\r\ny_correct = 1.644934066848226;\r\nassert(abs(y-y_correct)\u003c1e-13)\r\n\r\n%%\r\nn = 7;\r\ny = sumRecipPolyNum(n);\r\ny_correct = 1.32277925312239;\r\nassert(abs(y-y_correct)\u003c1e-13)\r\n\r\n%%\r\nn = 11;\r\ny = sumRecipPolyNum(n);\r\ny_correct = 1.19543411652963;\r\nassert(abs(y-y_correct)\u003c1e-13)\r\n\r\n%%\r\nn = 18;\r\ny = sumRecipPolyNum(n);\r\ny_correct = 1.11589671405633;\r\nassert(abs(y-y_correct)\u003c1e-13)\r\n\r\n%%\r\nn = 24:26;\r\ny = arrayfun(@sumRecipPolyNum,n);\r\ns = sum(y);\r\ns_correct = 3.247536806913290;\r\nfor k = 1:3\r\n str = num2str(y(k),'%1.15f');\r\n z(k) = str2num(flip(str(11:14)));\r\nend\r\nfs = sum(factor(sum(z)));\r\nfs_correct = 185;\r\nassert(abs(s-s_correct)\u003c1e-13)\r\nassert(isequal(fs,fs_correct))\r\n\r\n%%\r\nn = 31;\r\nindx = [4 5 7 9 10 11];\r\ns = num2str(sumRecipPolyNum(n),'%1.15f');\r\nd = num2str(sumRecipPolyNum(str2num(s(indx(randi(6))))),'%1.15f')-'0';\r\np = prod(d(3:13));\r\np_correct = 186624;\r\nassert(isequal(p,p_correct))\r\n\r\n%%\r\nn = 57;\r\ns = num2str(sumRecipPolyNum(n),'%1.15f');\r\nindx = [3 4; 4 10; 6 7; 7 10; 8 9; 8 10];\r\nfor k = size(indx,1):-1:1\r\n a(k) = str2num(s(indx(k,1):indx(k,2)));\r\nend\r\nassert(all(isprime(a)))","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2024-07-16T01:35:12.000Z","deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2024-07-16T01:34:17.000Z","updated_at":"2025-10-01T15:44:42.000Z","published_at":"2024-07-16T01:35:12.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAs explained in \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/60571\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eCody Problem 60571\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, a \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Polygonal_number\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003epolygonal number\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e is the number of dots arranged in the shape of a regular polygon. For example, 15 is a triangular number because dots can be arranged in the shape of a triangle with rows of 1, 2, 3, 4, and 5 dots. The number 16 is a square number because dots can be arranged in four rows of four. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to sum the reciprocals of polygonal numbers. In particular, compute\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y = sum[1/P_n,s,{n,1,inf}]\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey = \\\\sum_{n=1}^\\\\infty \\\\frac{1}{P_{n,s}}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"P_n,s\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eP_{n,s}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eth \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"s\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e-gonal number (i.e., the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eth number corresponding to a regular polygon with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"s\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e sides). \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":49743,"title":"Determine aquifer properties: unsteady pump test in a confined aquifer","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 714.15px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 357.075px; transform-origin: 407px 357.075px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 382.358px 7.79167px; transform-origin: 382.358px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAn important task in characterizing the flow of groundwater is to determine the properties of the aquifer, or the underground water-bearing formation. One approach is to disturb the aquifer, observe its response, and fit a theoretical formula to the observations. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 106.633px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 53.3167px; text-align: left; transform-origin: 384px 53.3167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 297.567px 7.79167px; transform-origin: 297.567px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor example, suppose a confined aquifer initially has no flow. In that case, the piezometric head \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eh\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 67.675px 7.79167px; transform-origin: 67.675px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, or the level to which water would rise in an observation well, would be a uniform value \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"h0\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 124.467px 7.79167px; transform-origin: 124.467px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. A well turned on and pumped at a rate \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"Q0\" style=\"width: 19px; height: 20px;\" width=\"19\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 38.9083px 7.79167px; transform-origin: 38.9083px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e will create a cone of depression; that is, it will draw down the piezometric head to a level \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"h(r)\" style=\"width: 29px; height: 18.5px;\" width=\"29\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24.8917px 7.79167px; transform-origin: 24.8917px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003er\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 95.2917px 7.79167px; transform-origin: 95.2917px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the radial distance from the well. Applying conservation of mass and Darcy’s law to this situation leads to a diffusion equation whose solution for the drawdown \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"s = h0 - h\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 79.3417px 7.79167px; transform-origin: 79.3417px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e as a function of distance \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003er\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 30.725px 7.79167px; transform-origin: 30.725px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and time \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003et\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7px 7.79167px; transform-origin: 7px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 44px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22px; text-align: left; transform-origin: 384px 22px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"s = (Q0/(4 pi T)) integral(exp(-x)/x, u, infinity)\" style=\"width: 117px; height: 44px;\" width=\"117\" height=\"44\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 7.79167px; transform-origin: 21.0083px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eT\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 65.9833px 7.79167px; transform-origin: 65.9833px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the transmissivity, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eS\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 67.15px 7.79167px; transform-origin: 67.15px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the storativity, and \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 36.05px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.025px; text-align: left; transform-origin: 384px 18.025px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"u = Sr^2/(4Tt)\" style=\"width: 49.5px; height: 36px;\" width=\"49.5\" height=\"36\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 349.917px 7.79167px; transform-origin: 349.917px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that achieves the objective of a pumping test: to determine the transmissivity and storativity from measurements of drawdown in time. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 347.467px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 173.733px; text-align: left; transform-origin: 384px 173.733px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" style=\"vertical-align: baseline;width: 497px;height: 342px\" 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data-image-state=\"image-loaded\" width=\"497\" height=\"342\"\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [T,S] = confinedPumpTest(t,s,Q0,r)\r\n % t = time, s = drawdown, Q0 = pumping rate, r = distance from well\r\n T = f1(t,s,Q0,r);\r\n S = f2(t,s,Q0,r);\r\nend","test_suite":"%%\r\nQ0 = 0.15; % Pumping rate (m3/s)\r\nr = 100; % Distance from well (m)\r\nt = [100 1000 10000 1e5 2e5]; % Time (s)\r\ns = [0.0402 1.236 3.664 6.276 7.05]; % Drawdown (m)\r\n[T,S] = confinedPumpTest(t,s,Q0,r);\r\nT_correct = 1.042e-2; % Transmissivity (m2/s)\r\nS_correct = 9.864e-4; % Storativity\r\nassert(abs(T-T_correct)/T_correct \u003c 1e-3 \u0026\u0026 abs(S-S_correct)/S_correct \u003c 1e-3)\r\n\r\n%%\r\nQ0 = 0.30; % Pumping rate (m3/s)\r\nr = 100; % Distance from well (m)\r\nt = [100 1000 10000 1e5 2e5]; % Time (s)\r\ns = [0.0804 2.472 7.328 12.552 14.1]; % Drawdown (m)\r\n[T,S] = confinedPumpTest(t,s,Q0,r);\r\nT_correct = 1.042e-2; % Transmissivity (m2/s)\r\nS_correct = 9.864e-4; % Storativity\r\nassert(abs(T-T_correct)/T_correct \u003c 1e-3 \u0026\u0026 abs(S-S_correct)/S_correct \u003c 1e-3)\r\n\r\n%%\r\nQ0 = 0.1; % Pumping rate (m3/s)\r\nr = 40; % Distance from well (m)\r\nt = [300 2000 8000 12000 24000 40000]; % Time (s)\r\ns = [0.494 1.749 2.830 3.153 3.709 4.120]; % Drawdown (m)\r\n[T,S] = confinedPumpTest(t,s,Q0,r);\r\nT_correct = 9.838e-3; % Transmissivity (m2/s)\r\nS_correct = 3.4e-3; % Storativity\r\nassert(abs(T-T_correct)/T_correct \u003c 1e-3 \u0026\u0026 abs(S-S_correct)/S_correct \u003c 1e-3)\r\n\r\n%%\r\nQ0 = 0.1; % Pumping rate (m3/s)\r\nr = 65; % Distance from well (m)\r\nt = [300 2000 8000 12000 24000 40000]; % Time (s)\r\ns = [0.125 1.050 2.067 2.383 2.931 3.339]; % Drawdown (m)\r\n[T,S] = confinedPumpTest(t,s,Q0,r);\r\nT_correct = 9.838e-3; % Transmissivity (m2/s)\r\nS_correct = 3.4e-3; % Storativity\r\nassert(abs(T-T_correct)/T_correct \u003c 2e-3 \u0026\u0026 abs(S-S_correct)/S_correct \u003c 1e-3)\r\n\r\n%%\r\nQ0 = 0.05; % Pumping rate (m3/s)\r\nr = 5+10*rand; % Distance from well (m)\r\nt = [4e5 9e5 14e5 19e5 24e5]; % Time (s)\r\ns = [0.859 0.918 0.951 0.973 0.991]; % Drawdown (m)\r\n[T,S] = confinedPumpTest(t,s,Q0,r);\r\nlogfit = polyfit(log(t),s,1); \r\nTapprox = Q0/(4*pi*logfit(1)); \r\nSapprox = 2.25*Tapprox*exp(-logfit(2)/logfit(1))/r^2; \r\nassert(abs(T-Tapprox)/Tapprox \u003c 1e-3 \u0026\u0026 abs(S-Sapprox)/Sapprox \u003c 2e-3)","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":4,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-01-04T00:26:53.000Z","updated_at":"2026-01-09T18:01:40.000Z","published_at":"2021-01-04T05:23:32.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAn important task in characterizing the flow of groundwater is to determine the properties of the aquifer, or the underground water-bearing formation. One approach is to disturb the aquifer, observe its response, and fit a theoretical formula to the observations. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example, suppose a confined aquifer initially has no flow. In that case, the piezometric head \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"h\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eh\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, or the level to which water would rise in an observation well, would be a uniform value \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"h0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eh_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. A well turned on and pumped at a rate \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Q0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eQ_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e will create a cone of depression; that is, it will draw down the piezometric head to a level \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"h(r)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eh(r)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"r\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003er\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the radial distance from the well. Applying conservation of mass and Darcy’s law to this situation leads to a diffusion equation whose solution for the drawdown \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"s = h0 - h\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es = h_0-h\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e as a function of distance \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"r\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003er\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and time \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"t\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003et\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"s = (Q0/(4 pi T)) integral(exp(-x)/x, u, infinity)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es = {Q_0 \\\\over 4 \\\\pi T} \\\\int_u^\\\\infty {e^{-x} \\\\over x} dx\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"T\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eT\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the transmissivity, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"S\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eS\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the storativity, and \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"u = Sr^2/(4Tt)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu = {S r^2 \\\\over 4 T t}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that achieves the objective of a pumping test: to determine the transmissivity and storativity from measurements of drawdown in time. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"342\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"497\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" 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\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":46012,"title":"Find the zeros of the Bessel function of the first kind","description":"\u003chttps://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html Bessel functions\u003e are important in many problems in mathematical physics--especially those with circular symmetry. Examples include the vibrations of a circular membrane and conduction of heat in a cylinder. Solving such problems usually requires finding the zeros of the Bessel functions and their derivatives. Like sine and cosine, the Bessel function of the first kind oscillates--as in the figure below. However, its zeros are not as easily predicted.\r\n\r\nFind the kth zero of the Bessel function of the first kind of order ν (nu) and its derivative. For Bessel functions with ν \u003e 0, skip the zero at z = 0. For the derivative of the Bessel function of order ν = 0, start counting with the zero at z = 0.\r\n\r\n\r\n\u003c\u003chttps://www.mathworks.com/help/examples/matlab/win64/PlotBesselFunctionsOfFirstKindExample_01.png\u003e\u003e","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 569.467px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 284.733px; transform-origin: 407px 284.733px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eBessel functions\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 305.617px 7.8px; transform-origin: 305.617px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are important in many problems in mathematical physics--especially those with circular symmetry. Examples include the vibrations of a circular membrane and conduction of heat in a cylinder. Solving such problems usually requires finding the zeros of the Bessel functions and their derivatives. Like sine and cosine, the Bessel function of the first kind oscillates--as in the figure below. However, its zeros are not as easily predicted.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 27.2333px 7.8px; transform-origin: 27.2333px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFind the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ek\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 165.683px 7.8px; transform-origin: 165.683px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eth zero of the Bessel function of the first kind of order \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eν\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 138.083px 7.8px; transform-origin: 138.083px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and its derivative. For Bessel functions with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eν\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 30.15px 7.8px; transform-origin: 30.15px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u0026gt; 0, skip the zero at \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ez\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 166.667px 7.8px; transform-origin: 166.667px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e = 0. For the derivative of the Bessel function of order \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eν\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 109.1px 7.8px; transform-origin: 109.1px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e = 0, start counting with the zero at \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ez\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.8167px 7.8px; transform-origin: 13.8167px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e = 0.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 425.467px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 212.733px; text-align: center; transform-origin: 384px 212.733px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" style=\"vertical-align: baseline\" 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\" data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [j,jp] = BesselJzero(nu,k)\r\n% k = number of the zero, nu = order of the Bessel function\r\n% j = zero of the Bessel function, jp = zero of the derivative of the Bessel function\r\n j = ...;\r\n jp = ...; \r\nend","test_suite":"%%\r\nnu = 0;\r\nk = 1; \r\nj_correct = 2.404825557695773;\r\nassert(abs(BesselJzero(nu,k)-j_correct)/j_correct \u003c 1e-4)\r\n\r\n%%\r\nnu = 1;\r\nk = 3; \r\nj_correct = 10.17346813506272;\r\nassert(abs(BesselJzero(nu,k)-j_correct)/j_correct \u003c 1e-4)\r\n\r\n%%\r\nnu = 2;\r\nk = 2; \r\nj_correct = 8.41724414039981;\r\njp_correct = 6.70613;\r\n[j,jp] = BesselJzero(nu,k);\r\nassert(abs(j-j_correct)/j_correct \u003c 1e-4)\r\nassert(abs(jp-jp_correct)/jp_correct \u003c 1e-4)\r\n\r\n%% \r\nnu = 0;\r\nj_correct = [2.404826 5.520078 8.653728 11.791534 14.930918];\r\njp_correct = [0 3.831706 7.015587 10.173468 13.323692];\r\nfor k = 1:5\r\n [j(k) jp(k)] = BesselJzero(nu,k);\r\nend\r\nassert(all(abs(j-j_correct)/j_correct \u003c 1e-4))\r\nassert(all(abs(jp-jp_correct)/jp_correct \u003c 1e-4))\r\n\r\n%% \r\nnu = 3;\r\nj_correct = [6.380162 9.761023 13.015201 16.223466 19.409415];\r\njp_correct = [4.20119 8.01524 11.3459 14.5858 17.7887];\r\nfor k = 1:5\r\n [j(k) jp(k)] = BesselJzero(nu,k);\r\nend\r\nassert(all(abs(j-j_correct)/j_correct \u003c 1e-4))\r\nassert(all(abs(jp-jp_correct)/jp_correct \u003c 1e-4))\r\n\r\n%%\r\nnu = 1/3;\r\nk = 5; \r\nj_correct = 15.450649;\r\nassert(abs(BesselJzero(nu,k)-j_correct)/j_correct \u003c 1e-4)\r\n\r\n%%\r\nnu = 1/2;\r\nk = randi(5);\r\nj_correct = k*pi;\r\nassert(abs(BesselJzero(nu,k)-j_correct)/j_correct \u003c 1e-4)","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":4,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-07-01T22:27:27.000Z","updated_at":"2020-07-30T20:57:30.000Z","published_at":"2020-07-02T02:25:18.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eBessel functions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e are important in many problems in mathematical physics--especially those with circular symmetry. Examples include the vibrations of a circular membrane and conduction of heat in a cylinder. Solving such problems usually requires finding the zeros of the Bessel functions and their derivatives. Like sine and cosine, the Bessel function of the first kind oscillates--as in the figure below. However, its zeros are not as easily predicted.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFind the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"k\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eth zero of the Bessel function of the first kind of order \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"nu\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\nu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and its derivative. For Bessel functions with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"nu\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\nu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e \u0026gt; 0, skip the zero at \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"z\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ez\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e = 0. For the derivative of the Bessel function of order \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"nu\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\nu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e = 0, start counting with the zero at \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"z\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ez\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e = 0.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc 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the generalized hypergeometric function","description":"The \u003chttps://en.wikipedia.org/wiki/Generalized_hypergeometric_function generalized hypergeometric function\u003e is defined as \r\n\r\n\u003c\u003chttps://wikimedia.org/api/rest_v1/media/math/render/svg/1622e60ecca4a7a8287805cbc798387110f49c68\u003e\u003e\r\n\r\n \r\n \r\nThe numbers _p_ and _q_ are the numbers of values _a_ and _b_ in the numerator and denominator (respectively), and the Pochhammer symbol (a)_n is defined by\r\n\r\n\u003c\u003chttps://wikimedia.org/api/rest_v1/media/math/render/svg/c560a95c630b385d8bdf14da55e36d1286d8c68f\u003e\u003e\r\n\r\n`\r\n\r\nMany other functions can be expressed in terms of the generalized hypergeometric function. For example, \r\n\r\n\r\n exp(x) = pFq([],[],x)\r\n cos(x) = pFq([],1/2,-x^2/4)\r\n besselj(0,x) = pFq([],1,-x^2/4)\r\n \r\nThe generalized hypergeometric function can be computed with |hypergeom| from the Symbolic Math Toolbox, but it is not available in Cody or basic MATLAB.\r\n\r\nWrite a function to evaluate the generalized hypergeometric function. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 395.9px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 197.95px; transform-origin: 407px 197.95px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12.0667px 7.8px; transform-origin: 12.0667px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Generalized_hypergeometric_function\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003egeneralized hypergeometric function\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 41.2333px 7.8px; transform-origin: 41.2333px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is defined as\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 45px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22.5px; text-align: left; transform-origin: 384px 22.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6px 7.8px; transform-origin: 13.6px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"(See the definition at the Wikipedia link)\" style=\"width: 301px; height: 45px;\" width=\"301\" height=\"45\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.8167px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.4167px; text-align: left; transform-origin: 384px 21.4167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 41.2333px 7.8px; transform-origin: 41.2333px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe numbers\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ep\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6167px 7.8px; transform-origin: 13.6167px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eq\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 82.85px 7.8px; transform-origin: 82.85px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are the numbers of values\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6167px 7.8px; transform-origin: 13.6167px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eb\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 180.1px 7.8px; transform-origin: 180.1px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in the numerator and denominator (respectively), and the Pochhammer symbol \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"(a)_n\" style=\"width: 27px; height: 20px;\" width=\"27\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 41.2333px 7.8px; transform-origin: 41.2333px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is defined by\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21.8167px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.9167px; text-align: left; transform-origin: 384px 10.9167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6px 7.8px; transform-origin: 13.6px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"(a)_n = a*(a+1)*(a+2)...(a+n-1)\" style=\"width: 184px; height: 20px;\" width=\"184\" height=\"20\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 327.917px 7.8px; transform-origin: 327.917px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eMany other functions can be expressed in terms of the generalized hypergeometric function. For example,\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21.8167px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.9167px; text-align: left; transform-origin: 384px 10.9167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6px 7.8px; transform-origin: 13.6px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"exp(x) = pFq([],[],x)\" style=\"width: 104.5px; height: 21px;\" width=\"104.5\" height=\"21\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 39px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 19.5px; text-align: left; transform-origin: 384px 19.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6px 7.8px; transform-origin: 13.6px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"cos(x0 = pFq([],1/2,-x^2/4)\" style=\"width: 160.5px; height: 39px;\" width=\"160.5\" height=\"39\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 39px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 19.5px; text-align: left; transform-origin: 384px 19.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6px 7.8px; transform-origin: 13.6px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"besselj(0,x) = pFq([],1,-x^2/4)\" style=\"width: 149px; height: 39px;\" width=\"149\" height=\"39\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.45px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.2333px; text-align: left; transform-origin: 384px 21.2333px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 379.9px 7.8px; transform-origin: 379.9px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere the dash means that the list of parameters is empty. The generalized hypergeometric function can be computed with\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 34.65px 7.8px; transform-origin: 34.65px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 34.65px 8.25px; transform-origin: 34.65px 8.25px; \"\u003ehypergeom\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 252.3px 7.8px; transform-origin: 252.3px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e from the Symbolic Math Toolbox, but it is not available in Cody or basic MATLAB.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 213.05px 7.8px; transform-origin: 213.05px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to evaluate the generalized hypergeometric function.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = pFq(a,b,z)\r\n y = f(a,b,z);\r\nend","test_suite":"%% exp(x)\r\na = [];\r\nb = [];\r\nz = 1;\r\npFq_correct = exp(z);\r\nassert(abs(pFq(a,b,z)-pFq_correct)/pFq_correct \u003c 1e-8)\r\n\r\n%% cos(x)\r\na = [];\r\nb = 1/2;\r\nx = pi/4;\r\nz = -x^2/4;\r\npFq_correct = 1/sqrt(2);\r\nassert(abs(pFq(a,b,z)-pFq_correct)/pFq_correct \u003c 1e-8)\r\n\r\n%% J_0(x)\r\na = [];\r\nb = 1;\r\nx = 1;\r\nz = -x^2/4;\r\npFq_correct = besselj(0,x);\r\nassert(abs(pFq(a,b,z)-pFq_correct)/pFq_correct \u003c 1e-8)\r\n\r\n%% 1/(1-x)^a\r\na = 2;\r\nb = [];\r\nz = 1/2;\r\npFq_correct = 4;\r\nassert(abs(pFq(a,b,z)-pFq_correct)/pFq_correct \u003c 1e-8)\r\n\r\n%% Example from \"help hypergeom\"--current version of help gives hypergeom(1,2,3) = exp(1)-1\r\na = 1;\r\nb = 2;\r\nz = 3;\r\npFq_correct = (exp(3)-1)/3;\r\nassert(abs(pFq(a,b,z)-pFq_correct)/pFq_correct \u003c 1e-8)\r\n\r\n%% Hypergeometric function F(a,b; c; x)\r\na = [1 2];\r\nb = 4;\r\nz = 0.2;\r\npFq_correct = 1.113869211474147;\r\nassert(abs(pFq(a,b,z)-pFq_correct)/pFq_correct \u003c 1e-8)\r\n\r\n%% \r\na = [1 1];\r\nb = 2;\r\nz = rand;\r\npFq_correct = -log(1-z)/z;\r\nassert(abs(pFq(a,b,z)-pFq_correct)/pFq_correct \u003c 1e-8)\r\n\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":6,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-07-11T20:25:32.000Z","updated_at":"2026-01-09T17:23:16.000Z","published_at":"2020-07-11T22:27:35.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Generalized_hypergeometric_function\\\"\u003e\u003cw:r\u003e\u003cw:t\u003egeneralized hypergeometric function\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e is defined as\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"(See the definition at the Wikipedia link)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e_pF_q(a_1,\\\\ldots,a_p; c_1,\\\\ldots,c_q; x) = \\\\sum_{n=0}^\\\\infty \\\\frac{(a_1)_n\\\\cdots (a_p)_n}{(c_1)_n\\\\cdots (c_q)_n}\\\\frac{x^n}{n!}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe numbers\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"q\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eq\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e are the numbers of values\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"b\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eb\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e in the numerator and denominator (respectively), and the Pochhammer symbol \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"(a)_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e(a)_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is defined by\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"(a)_n = a*(a+1)*(a+2)...(a+n-1)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e(a)_n = a(a+1)\\\\cdots (a+n-1)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eMany other functions can be expressed in terms of the generalized hypergeometric function. For example,\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"exp(x) = pFq([],[],x)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ee^x = {}_0F_0(-,-,x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"cos(x0 = pFq([],1/2,-x^2/4)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\cos(x) = {}_0F_1\\\\left(-,\\\\frac{1}{2},-\\\\frac{x^2}{4}\\\\right)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"besselj(0,x) = pFq([],1,-x^2/4)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eJ_0(x) = {}_0F_1\\\\left(-,1,-\\\\frac{x^2}{4}\\\\right)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere the dash means that the list of parameters is empty. The generalized hypergeometric function can be computed with\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ehypergeom\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e from the Symbolic Math Toolbox, but it is not available in Cody or basic MATLAB.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to evaluate the generalized hypergeometric function.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":46066,"title":"Evaluate the logarithmic integral","description":"The \u003chttps://en.wikipedia.org/wiki/Logarithmic_integral_function logarithmic integral\u003e li(x) plays a role in number theory because the related function Li(x) = li(x) - li(2) provides an estimate for the number of primes less than x. MATLAB's Symbolic Toolbox has the function \u003chttps://www.mathworks.com/help/symbolic/sym.logint.html |logint|\u003e, but it is not available in basic MATLAB or Cody. \r\n\r\nWrite a function to evaluate the logarithmic integral.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 123px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 61.5px; transform-origin: 407px 61.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12.0667px 7.91667px; transform-origin: 12.0667px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Logarithmic_integral_function\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003elogarithmic integral\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"li(x)\" style=\"width: 30.5px; height: 19px;\" width=\"30.5\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 184px 7.91667px; transform-origin: 184px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e plays a role in number theory because the related function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPwAAAAmCAYAAADgDvYuAAAFLElEQVR4nO2dXbXrIBCFt4c4iIEYiIIqqIM6qINaiIZKqIdaqIZaOOeBzuq0DTDAQH7WfGvlpbmHJDtsmAHCBQzDMAzDMAzDMAzDMAzDMIQcX4cWE4Besby9YXrX5wJgUCqrf5XXlRRyBHCOHDllnhJv7AhXYTTpAdyhJ/gSDHBa3gFcPf/G9C5jhDPSA07Hb7rX76mN4yT4mxFvn50Q126E0zjb9AOAw6uQP3Y8X7+PieUdWRmXhL+5JV5HygD3IrfW8/RwBn/greec4U3vfEY4HZ54azhn+ImdPwjLnhDuLEd8vlt+XBE29BHOr0X0XxeVPtjczaRUwAFO8JoVREWgheDvRcPwpvcvXEMNw1NE5mNk5d3w2+jQ7yHTT5A38F74BUvCMsoNJWHHA3lpQyq3RtepQcjwgOldCjdgKKSXmJ0aaF9k3MG9gxt+G90j4tEGL+cZuI4ILcNLoQcsGoQQcoB7ri2Emt/EDC/F9J4nZvgUJoTTpSOc4X3vgEcbsbSLxh2yaW34BxTCkhVfTwstw5ve82gZnnr3UCRwi5wH3r38U+F6QVoannqAopAkkQviIq4RDcOb3n60DH95lRFCMptyg8zwQHj2JopmDn+N3AgNhKTQwVXcM34rLuVZobxxiUqvgSSHN73zkebwd895gnLzUsjwEiNLGhkvJYbv4SoVH3QI3XCKODRFxUczeR404PO6vrlPCoFyBpN6xNcrSI6cyj+n5971bonP8COcDn+e8xzNZyVtJaE65fxZjapGDz9AVgFzQ9QzK3/Ee9STepu50U+N6/JKUXLkhIwhPfeqd0tiPfwpcp6XUTroR+9Turim6LpaIX2sAtJN5rSGvIKf4CpcysM+kRd2afXwObrG9Nyj3i2JGV6S41PDWJq+UOolLYfeT1ZksQXDA5+jmKnLQylM3RJLGh5orzdfbtoidVqL4cm8qTMb2VHUVgzP86rUeV4aENkSSxu+td48jWiROq3B8B3yR9z/kBlFbcXwPKdKXURihk+ntd4aPXxK/V2D4WnBTs6iqN0bnq9tTv2CKXfqZG2j9Cnnt6h3S5Y2fInZi2YHcgzf4feFxipgh7xcBXgvTaS8MjWnzG0N1zpKLzm/Rb1bomF4WnOQuuqtxOz83poZ/ozfL4NiFRDIG72lTy/5/GjKWmKq+EvOwy8xSg9sT++WaBg+Z7ScpjVDZj8gXGeKFjelGp4WYHw/pKQCXpE2ekuDGhRS8rySBpKuCLew9OKy1x4vhIbhTW8/GoYHnL7SQTf6kCk0CEr+CjUIlEokRwip38PzT/m+WxdJBaQVQqEHvuA973vHZ0jK54cvcKFRLNykZYgtvhbTgnrJUsOb3n6olyw1/ARZBETvgsY3fIckhbojY++BgV2AH6EboYO3QB0+p1Ro15w5qCKHBOTXmavMsd1gvrlDf1unmtDyWZ+epnc5Az53e/reoouWGfvOc6jhCEXH/PNXyREK1emdJg2kSva0Cx1UwUI5rq+SxVpEmp4JrdUOnedQD7X2DzmImJ6mdzmh6b+D4PwcD4QbuRRvxQZ4T2i3v4EKPRR27RAyYf1rumtjetfngHYm9G26uWpOKNy1Q8CI+vu4bQXTuz5X1E9l5mbHNsOEejuj0Nddax8pbonpXRea4ailQYuNSKtzg+5/ikBcK5W7dUzvutTao58ak63t/T/LBN3KclEub2+Y3nWhVaha5uzgGtRdmJ3QHOzYdMjTCNO7Pload4plGYZhGIZhGIZhGIZhGIaxFP/Ka2gIYQHNwgAAAABJRU5ErkJggg==\" alt=\"Li(x) = li(x) - li(2)\" style=\"width: 126px; height: 19px;\" width=\"126\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.0667px 7.91667px; transform-origin: 40.0667px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e provides an estimate for the number of primes less than x. MATLAB's Symbolic Toolbox has the function\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/help/symbolic/sym.logint.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003elogint\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 75px 7.91667px; transform-origin: 75px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, but it is not available in basic MATLAB or Cody.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 158.967px 7.91667px; transform-origin: 158.967px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to evaluate the logarithmic integral.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 29.1833px 7.91667px; transform-origin: 29.1833px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSee also \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46081-set-soldner-s-constant\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problem 46081 \"Set Soldner's constant\"\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = logarithmicIntegral(x)\r\n y = f(x);\r\nend","test_suite":"%%\r\nx = 0;\r\ny_correct = 0;\r\nassert(abs(logarithmicIntegral(x)-y_correct)\u003c1e-8)\r\n\r\n%%\r\nx = 0.2;\r\ny_correct = -0.085126486728794;\r\nassert(abs(logarithmicIntegral(x)-y_correct)\u003c1e-8)\r\n\r\n%%\r\nx = 0.5;\r\ny_correct = -0.378671043061088;\r\nassert(abs(logarithmicIntegral(x)-y_correct)\u003c1e-8)\r\n\r\n%%\r\nx = 1;\r\ny = logarithmicIntegral(x);\r\nassert(isinf(y) \u0026\u0026 sign(y) == -1)\r\n\r\n%%\r\nx = 2;\r\ny_correct = 1.045163780117493;\r\nassert(abs(logarithmicIntegral(x)-y_correct)\u003c1e-8)\r\n\r\n%%\r\nx = 5;\r\ny_correct = 3.634588310032652;\r\nassert(abs(logarithmicIntegral(x)-y_correct)\u003c1e-8)\r\n\r\n%%\r\nx = 8;\r\ny_correct = 5.253718299558931;\r\nassert(abs(logarithmicIntegral(x)-y_correct)\u003c1e-8)\r\n\r\n%%\r\nx = 3+2i;\r\ny_correct = 2.558790740400258 + 1.594445119241119i;\r\ny = logarithmicIntegral(x);\r\nassert(abs(real(y)-real(y_correct))\u003c1e-8 \u0026\u0026 abs(imag(y)-imag(y_correct))\u003c1e-8)\r\n\r\n%%\r\nx = 3-0.2i;\r\ny_correct = 2.169086896211800 - 0.181703645882027i;\r\ny = logarithmicIntegral(x);\r\nassert(abs(real(y)-real(y_correct))\u003c1e-8 \u0026\u0026 abs(imag(y)-imag(y_correct))\u003c1e-8)\r\n\r\n%%\r\nx = -0.3-2i;\r\ny_correct = 0.999726888286245 - 3.238096925989443i;\r\ny = logarithmicIntegral(x);\r\nassert(abs(real(y)-real(y_correct))\u003c1e-8 \u0026\u0026 abs(imag(y)-imag(y_correct))\u003c1e-8)\r\n\r\n%%\r\nx = -5-3i;\r\ny_correct = 0.500720609942772 - 5.020802115037742i;\r\ny = logarithmicIntegral(x);\r\nassert(abs(real(y)-real(y_correct))\u003c1e-8 \u0026\u0026 abs(imag(y)-imag(y_correct))\u003c1e-8)\r\n\r\n%%\r\nx = 0.3i;\r\ny_correct = 0.074754684076440 + 3.053886180297906i;\r\ny = logarithmicIntegral(x);\r\nassert(abs(real(y)-real(y_correct))\u003c1e-8 \u0026\u0026 abs(imag(y)-imag(y_correct))\u003c1e-8)\r\n \r\n%%\r\nx = 0.4-2i;\r\ny_correct = 1.171012435933119 - 2.861745062394908i;\r\ny = logarithmicIntegral(x);\r\nassert(abs(real(y)-real(y_correct))\u003c1e-8 \u0026\u0026 abs(imag(y)-imag(y_correct))\u003c1e-8)","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":16,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-07-28T15:11:20.000Z","updated_at":"2026-01-29T16:19:49.000Z","published_at":"2020-07-29T01:46:40.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Logarithmic_integral_function\\\"\u003e\u003cw:r\u003e\u003cw:t\u003elogarithmic integral\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"li(x)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e{\\\\rm li}(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e plays a role in number theory because the related function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Li(x) = li(x) - li(2)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e{\\\\rm Li}(x) = {\\\\rm li}(x) - {\\\\rm li}(2)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e provides an estimate for the number of primes less than x. MATLAB's Symbolic Toolbox has the function\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/help/symbolic/sym.logint.html\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003elogint\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, but it is not available in basic MATLAB or Cody.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to evaluate the logarithmic integral.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSee also \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46081-set-soldner-s-constant\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem 46081 \\\"Set Soldner's constant\\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":54610,"title":"Evaluate the Kelvin functions","description":"The Kelvin functions ber, bei, ker, and kei are related to Bessel functions of order . When the order is not specified, the default is . The functions ker() and kei() appear in the solution for velocity in the boundary layer under water waves. \r\nWrite a function to compute the four Kelvin functions. Allow for a variable number of outputs. If the order is not specified, take it to be zero. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 114.75px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 57.375px; transform-origin: 407px 57.375px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63.75px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.875px; text-align: left; transform-origin: 384px 31.875px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 75.075px 7.50833px; transform-origin: 75.075px 7.50833px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe Kelvin functions ber\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"_nu(x)\" style=\"width: 27px; height: 20px;\" width=\"27\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.225px 7.50833px; transform-origin: 13.225px 7.50833px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, bei\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADYAAAAoCAYAAAC1mQk2AAADWUlEQVRoQ+2YO89NQRSGz/cDRFwqlbgkOgqiQaFASESiQLRCKFRCoZS4hehQKBQSvohGIShIUBAkVAqXjsol4gfwPjIrmUxmZs+cPWeTL2cnb07Onpk16123WbNnRnP0mZmjvEZTYpWe3a75n4W3leti05fp5SrhXo2sSXhsrxTYJBypUaRj7nGNL62R2ZoYpHYJ/LZ+LjuBRQZrSWyDNr4rrBS+tWbl5N3X73XhVpf8VsQWaaP3LlQ6N+1SKjO+WmOPhbXCp5ycVsTIgYPCih5Kly4lJCko2yZNbChvGQ9C/qmwRkhW3RYeO6QNrgqLJ5hboXM+6MVDF/pRx7Ug9tJJXlcaSw3mEY5U3oUpWX2JEYZfhdPCyQYKl4qwKEmGY19iFu8npNH5Uq3cPArAEuFdJISR+yuTQ7bvPs2JVuG+xMxyG7XBswJilGvWbBGWu/m+txm/48Z+6DdV1jsN2pcYZf6cUEoML/10HqIAQO6jsF6YL7wSrgh7BPInmUMa++3mRjuRoYn5TjWj8A6Cs8LFVGhFogFiD4ToefYviVk4oTOeonzXFKD/lhiEUI6HfMqFXeiwzmrc12OcJTeF0hwLFaSp3eo8VnMOTrx4dG4QyQ3/FaWaQoHHam4Ftu8OrYteQPt6zMKJSlZ0T/JY2VFhr5JKRoxjhYeiE+3yWxDD6mxQE0qcV2/cOso9T033wp6cccnbRIqYfWdgQ9/VJC3njW8ly7Ok9QKLI+OFcNTJjuUZveCFlDf0nqKT7XZixAiRM8KCyGK78vttjF1bzmp+qq1CJs9z4Zpw25sbnmfHNEbOpUp/kSFTHjMrhk0mnfwBIbwH5S6a/nkFOQ5i/5uIhaU5NhwPHD5Ct9cZ4n/nx4jZGUHs+zGMgpeEVC5B2veEKYS8UwLh/SjhVQyzOTNusvisd0PorKAxYubqMJlRPNfyWAO7O+LR0Orj/LeQ36nFnQ13jBiJe1jwyy/v8FRX5TOv0r+1/lJV/IUqFYrfnTlxNw9hxDUDZL8Mufl4jo5/f0NyGPaJUPwFLPSYJbI1pXgOorXhRT7NaxSSY8kKifmll+KR/WAyTqIMtSYkZtf1L4VhN5Se1fu0aKmqNx1iwZTYEFZuucfUYy2tOYSsqceGsHLLPf4AnTK3KUKaO2cAAAAASUVORK5CYII=\" alt=\"_nu(x)\" style=\"width: 27px; height: 20px;\" width=\"27\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6083px 7.50833px; transform-origin: 13.6083px 7.50833px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, ker\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"_nu(x)\" style=\"width: 27px; height: 20px;\" width=\"27\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 26.45px 7.50833px; transform-origin: 26.45px 7.50833px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and kei\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADYAAAAoCAYAAAC1mQk2AAADWUlEQVRoQ+2YO89NQRSGz/cDRFwqlbgkOgqiQaFASESiQLRCKFRCoZS4hehQKBQSvohGIShIUBAkVAqXjsol4gfwPjIrmUxmZs+cPWeTL2cnb07Onpk16123WbNnRnP0mZmjvEZTYpWe3a75n4W3leti05fp5SrhXo2sSXhsrxTYJBypUaRj7nGNL62R2ZoYpHYJ/LZ+LjuBRQZrSWyDNr4rrBS+tWbl5N3X73XhVpf8VsQWaaP3LlQ6N+1SKjO+WmOPhbXCp5ycVsTIgYPCih5Kly4lJCko2yZNbChvGQ9C/qmwRkhW3RYeO6QNrgqLJ5hboXM+6MVDF/pRx7Ug9tJJXlcaSw3mEY5U3oUpWX2JEYZfhdPCyQYKl4qwKEmGY19iFu8npNH5Uq3cPArAEuFdJISR+yuTQ7bvPs2JVuG+xMxyG7XBswJilGvWbBGWu/m+txm/48Z+6DdV1jsN2pcYZf6cUEoML/10HqIAQO6jsF6YL7wSrgh7BPInmUMa++3mRjuRoYn5TjWj8A6Cs8LFVGhFogFiD4ToefYviVk4oTOeonzXFKD/lhiEUI6HfMqFXeiwzmrc12OcJTeF0hwLFaSp3eo8VnMOTrx4dG4QyQ3/FaWaQoHHam4Ftu8OrYteQPt6zMKJSlZ0T/JY2VFhr5JKRoxjhYeiE+3yWxDD6mxQE0qcV2/cOso9T033wp6cccnbRIqYfWdgQ9/VJC3njW8ly7Ok9QKLI+OFcNTJjuUZveCFlDf0nqKT7XZixAiRM8KCyGK78vttjF1bzmp+qq1CJs9z4Zpw25sbnmfHNEbOpUp/kSFTHjMrhk0mnfwBIbwH5S6a/nkFOQ5i/5uIhaU5NhwPHD5Ct9cZ4n/nx4jZGUHs+zGMgpeEVC5B2veEKYS8UwLh/SjhVQyzOTNusvisd0PorKAxYubqMJlRPNfyWAO7O+LR0Orj/LeQ36nFnQ13jBiJe1jwyy/v8FRX5TOv0r+1/lJV/IUqFYrfnTlxNw9hxDUDZL8Mufl4jo5/f0NyGPaJUPwFLPSYJbI1pXgOorXhRT7NaxSSY8kKifmll+KR/WAyTqIMtSYkZtf1L4VhN5Se1fu0aKmqNx1iwZTYEFZuucfUYy2tOYSsqceGsHLLPf4AnTK3KUKaO2cAAAAASUVORK5CYII=\" alt=\"_nu(x)\" style=\"width: 27px; height: 20px;\" width=\"27\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 124.075px 7.50833px; transform-origin: 124.075px 7.50833px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are related to Bessel functions of order \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eν\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 72.725px 7.50833px; transform-origin: 72.725px 7.50833px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. When the order is not specified, the default is \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"nu = 0\" style=\"width: 36px; height: 18px;\" width=\"36\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 59.8917px 7.50833px; transform-origin: 59.8917px 7.50833px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The functions ker(\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 29.175px 7.50833px; transform-origin: 29.175px 7.50833px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e) and kei(\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 175.825px 7.50833px; transform-origin: 175.825px 7.50833px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e) appear in the solution for velocity in the boundary layer under water waves. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 372.867px 7.50833px; transform-origin: 372.867px 7.50833px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to compute the four Kelvin functions. Allow for a variable number of outputs. If the order is not specified, take it to be zero. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [ber,bei,ker,kei] = kelvin(x,nu)\r\n ber = besselr(nu,x);\r\n bei = besseli(nu,x);\r\n ker = kesselr(nu,x);\r\n kei = kesseli(nu,x);\r\nend","test_suite":"%%\r\nx = 3;\r\ntol = 1e-15;\r\n[ber,bei,ker,kei] = kelvin(x);\r\n[ber_correct,bei_correct,ker_correct,kei_correct] = deal(-0.221380249598694,1.937586785266043,-0.067029233303799,-0.051121884045987);\r\nassert(abs(ber-ber_correct)\u003ctol \u0026 abs(bei-bei_correct)\u003ctol \u0026 abs(ker-ker_correct)\u003ctol \u0026 abs(kei-kei_correct)\u003ctol)\r\n\r\n%%\r\nx = 1:4;\r\ntol = 1e-15;\r\nber = kelvin(x);\r\nber_correct = [0.984381781213087 0.7517341827138085 -0.221380249598694 -2.563416557258581];\r\nassert(all(abs(ber-ber_correct)\u003ctol))\r\n\r\n%%\r\nx = -0.4;\r\nnu = 1;\r\ntol = 1e-15;\r\n[ber,bei] = kelvin(x,nu);\r\n[ber_correct,bei_correct] = deal(0.1442308644531633,-0.1385741359112079);\r\nassert(abs(ber-ber_correct)\u003ctol \u0026 abs(bei-bei_correct)\u003ctol)\r\n\r\n%%\r\nx = 3.4;\r\nnu = 0.5;\r\ntol = 1e-15;\r\n[ber,bei,ker,kei] = kelvin(x,nu);\r\n[ber_correct,bei_correct,ker_correct,kei_correct] = deal(-2.245652084214816,0.82701468622679,-0.05553905648843065,0.02619201937598225);\r\nassert(abs(ber-ber_correct)\u003ctol \u0026 abs(bei-bei_correct)\u003ctol \u0026 abs(ker-ker_correct)\u003ctol \u0026 abs(kei-kei_correct)\u003ctol)\r\n\r\n%%\r\nx = [-psi(1) exp(1) pi];\r\nnu = 1/3;\r\ntol = 2e-14;\r\n[ber,bei,ker,kei] = kelvin(x,nu);\r\nber_correct = [0.4900252831480887 -0.7214812981316725 -1.432808723213069];\r\nbei_correct = [0.5553968849316957 1.532550247673432 1.535388618042967];\r\nker_correct = [0.2909569660163571 -0.1038243609935959 -0.07552171571938433];\r\nkei_correct = [-0.983688891425642 -0.03413110664720041 -0.001259453778330345];\r\nassert(all(abs(ber-ber_correct)\u003ctol) \u0026 all(abs(bei-bei_correct)\u003ctol) \u0026 all(abs(ker-ker_correct)\u003ctol) \u0026 all(abs(kei-kei_correct)\u003ctol))\r\n\r\n%%\r\nfiletext = fileread('kelvin.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'regexp') || contains(filetext, 'import'); \r\nassert(~illegal)","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2022-05-06T01:51:35.000Z","deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2022-05-06T01:49:11.000Z","updated_at":"2026-01-09T20:00:53.000Z","published_at":"2022-05-06T01:51:35.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe Kelvin functions ber\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"_nu(x)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e_\\\\nu(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, bei\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"_nu(x)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e_\\\\nu(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, ker\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"_nu(x)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e_\\\\nu(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and kei\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"_nu(x)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e_\\\\nu(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e are related to Bessel functions of order \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"nu\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\nu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. When the order is not specified, the default is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"nu = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\nu = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The functions ker(\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e) and kei(\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e) appear in the solution for velocity in the boundary layer under water waves. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to compute the four Kelvin functions. Allow for a variable number of outputs. If the order is not specified, take it to be zero. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":45997,"title":"Evaluate the zeta function for complex arguments","description":"\u003chttps://www.mathworks.com/matlabcentral/cody/problems/45988 Cody Problem 45988\u003e involved computing the Riemann zeta function for real arguments greater than 1. Code that works for that problem can reveal the connection between pi and the values of the zeta function evaluated at positive even integers; this connection is explored in \u003chttps://www.mathworks.com/matlabcentral/cody/problems/45939-estimate-pi-from-certain-values-of-the-zeta-function Cody Problem 45939\u003e. However, to test the Riemann hypothesis--that all non-trivial zeros of the zeta function have a real part of 1/2, one needs to compute the zeta function for complex arguments.\r\n\r\nWrite a function to evaluate the zeta function for complex arguments.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 114px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 57px; transform-origin: 407px 57px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/45988\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problem 45988\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 315.85px 7.8px; transform-origin: 315.85px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e involved computing the Riemann zeta function for real arguments greater than 1. Code that works for that problem can reveal the connection between \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eπ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 222.1px 7.8px; transform-origin: 222.1px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and the values of the zeta function evaluated at positive even integers; this connection is explored in\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/45939-estimate-pi-from-certain-values-of-the-zeta-function\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problem 45939\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 217.433px 7.8px; transform-origin: 217.433px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. However, to test the Riemann hypothesis--that all non-trivial zeros of the zeta function have a real part of 1/2, one needs to compute the zeta function for complex arguments.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 213.017px 7.8px; transform-origin: 213.017px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to evaluate the zeta function for complex arguments.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function z = zeta2(s)\r\n z = f(s);\r\nend","test_suite":"%%\r\ns = 2;\r\nz_correct = pi^2/6;\r\nassert(abs(zeta2(s)-z_correct)/z_correct \u003c 1e-8)\r\n\r\n%%\r\ns = 1;\r\nassert(isinf(zeta2(s)))\r\n\r\n%%\r\ns = 1/2;\r\nz_correct = -1.460354508809587;\r\nassert(abs((zeta2(s)-z_correct)/z_correct) \u003c 1e-8)\r\n\r\n%%\r\ns = 0;\r\nz_correct = -0.5;\r\nassert(abs((zeta2(s)-z_correct)/z_correct) \u003c 1e-8)\r\n\r\n%%\r\ns = -1;\r\nz_correct = -1/12;\r\nassert(abs((zeta2(s)-z_correct)/z_correct) \u003c 1e-8)\r\n\r\n%%\r\ns = -2;\r\nz_correct = 0;\r\nassert(abs(zeta2(s)) \u003c 1e-12)\r\n\r\n%%\r\ns = 3+2*i;\r\nz_correct = 0.973041960418942 - 0.147695593000454i;\r\nassert(abs(real(zeta2(s))-real(z_correct))/real(z_correct) \u003c 1e-8) \r\nassert(abs((imag(zeta2(s))-imag(z_correct))/imag(z_correct)) \u003c 1e-8)\r\n\r\n%%\r\ns = -1+2*i;\r\nz_correct = 0.168915669770846 - 0.070515988908259i;\r\nassert(abs((real(zeta2(s))-real(z_correct))/real(z_correct)) \u003c 1e-8) \r\nassert(abs((imag(zeta2(s))-imag(z_correct))/imag(z_correct)) \u003c 1e-8)\r\n\r\n%%\r\ns = 0.75-3*i;\r\nz_correct = 0.580900396083837 + 0.095281202690117i;\r\nassert(abs(real(zeta2(s))-real(z_correct))/real(z_correct) \u003c 1e-8) \r\nassert(abs((imag(zeta2(s))-imag(z_correct))/imag(z_correct)) \u003c 1e-8)\r\n\r\n%%\r\ns = 5+2*i;\r\nz_correct = 1.001916538615071 - 0.034217062736354i;\r\nassert(abs((real(zeta2(s))-real(z_correct))/real(z_correct)) \u003c 1e-8) \r\nassert(abs((imag(zeta2(s))-imag(z_correct))/imag(z_correct)) \u003c 1e-8)\r\n\r\n%%\r\ns = 0.5+14.13472514173469379*i;\r\nassert(abs(real(zeta2(s))) \u003c 1e-12) \r\nassert(abs(imag(zeta2(s))) \u003c 1e-12)\r\n\r\n%%\r\ns = 0.5+21*i;\r\nz_correct = -0.005162064638102 - 0.024546964575122i;\r\nassert(abs((real(zeta2(s))-real(z_correct))/real(z_correct)) \u003c 1e-8) \r\nassert(abs((imag(zeta2(s))-imag(z_correct))/imag(z_correct)) \u003c 1e-8)\r\n\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":3,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":12,"test_suite_updated_at":"2020-06-29T02:07:46.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2020-06-28T04:41:50.000Z","updated_at":"2026-01-09T13:36:37.000Z","published_at":"2020-06-28T05:14:18.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/45988\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem 45988\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e involved computing the Riemann zeta function for real arguments greater than 1. Code that works for that problem can reveal the connection between \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"pi\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and the values of the zeta function evaluated at positive even integers; this connection is explored in\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/45939-estimate-pi-from-certain-values-of-the-zeta-function\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem 45939\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. However, to test the Riemann hypothesis--that all non-trivial zeros of the zeta function have a real part of 1/2, one needs to compute the zeta function for complex arguments.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to evaluate the zeta function for complex arguments.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":59521,"title":"Integrate a power tower","description":"Write a function to compute this integral\r\n\r\nwhere . That is, the integrand is (x to the x) to the (x to the x) to the (x to the x)...","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 104px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 52px; transform-origin: 407px 52px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 122.783px 8px; transform-origin: 122.783px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to compute this integral\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 44px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22px; text-align: left; transform-origin: 384px 22px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg 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AnGgx7jdhPsh4iVErvgcu4lMytGj7v56uakInAJxxMhe0XTq5tTcDXV+3D7Fyp+vuWVtajvpOSHQFAKnQBzxwiL39PSVRMkR+jEHMJzAbjWJ95g01cCqjBBYA4FTIA62KlyL6PeA+GVCd9p36R2afRi7y/k99qNfCeBkpNXGGr1SeTaPwCkQB43A4IckLptcNbnLxK+8K2kkzrNw6zuH27jR+9Emd1TmUVKO0giBXSBwKsSxi8ZQJYXAXhAQceylpVRPIdAQAiKOhhpDVRECe0FAxLGXllI9hUBDCIg4GmoMVUUI7AUBEcdeWkr1FAINISDiaKgxVBUhsBcE/g+tfmSG+LdlUAAAAABJRU5ErkJggg==\" width=\"135\" height=\"44\" alt=\"I = integral((x^x)^(x^x)^(x^x)...,{x,a,0})\" style=\"width: 135px; height: 44px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 8px; transform-origin: 21.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"63\" height=\"18\" style=\"width: 63px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 226.3px 8px; transform-origin: 226.3px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. That is, the integrand is (x to the x) to the (x to the x) to the (x to the x)...\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function I = intPowerTower(a)\r\n I = integral(x^x^x^x^x^x^x^x,0,a);\r\nend","test_suite":"%%\r\na = 0;\r\nI = intPowerTower(a);\r\nassert(abs(I)\u003c1e-6)\r\n\r\n%%\r\na = 1/100;\r\nI = intPowerTower(a);\r\nI_correct = 0.00975627404012066;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 1/20;\r\nI = intPowerTower(a);\r\nI_correct = 0.04621245261821598;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 1/10;\r\nI = intPowerTower(a);\r\nI_correct = 0.0886781687569094;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 1/5;\r\nI = intPowerTower(a);\r\nI_correct = 0.1685639964895788;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 1/4;\r\nI = intPowerTower(a);\r\nI_correct = 0.2071658901263798;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 3/8;\r\nI = intPowerTower(a);\r\nI_correct = 0.30215124860335973;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 1/2;\r\nI = intPowerTower(a);\r\nI_correct = 0.3972053202401857;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 2/3;\r\nI = intPowerTower(a);\r\nI_correct = 0.5277402852630483;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 3/4;\r\nI = intPowerTower(a);\r\nI_correct = 0.5959989560650945;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 5/6;\r\nI = intPowerTower(a);\r\nI_correct = 0.6671963910854818;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 1;\r\nI = intPowerTower(a);\r\nI_correct = 0.822467033424113;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = (rand+3)/4;\r\nI = intPowerTower(a);\r\nI_correct = polyval([0.3875275 -0.9886411 1.132527 0.1505356 0.1405179],a);\r\nassert(abs(I-I_correct)\u003c5e-6)\r\n\r\n%%\r\nfiletext = fileread('intPowerTower.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert') || contains(filetext,'regexp') || contains(filetext,'find') || contains(filetext,'switch'); \r\nassert(~illegal)\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":3,"created_by":46909,"edited_by":46909,"edited_at":"2024-01-03T15:06:22.000Z","deleted_by":null,"deleted_at":null,"solvers_count":7,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-12-31T18:50:11.000Z","updated_at":"2026-01-28T06:58:04.000Z","published_at":"2023-12-31T18:50:21.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to compute this integral\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"I = integral((x^x)^(x^x)^(x^x)...,{x,a,0})\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eI = \\\\int_0^a {(x^x)^{(x^x)^{(x^x)\\\\ldots}} dx\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e0 \\\\le a \\\\le 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. That is, the integrand is (x to the x) to the (x to the x) to the (x to the x)...\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51783,"title":"Solve an ODE: equation B","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 213.25px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 106.625px; transform-origin: 407px 106.625px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 211.358px 7.91667px; transform-origin: 211.358px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve the following ordinary differential equation: \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 39px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 19.5px; text-align: left; transform-origin: 384px 19.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg src=\"data:image/png;base64,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alt=\"y\u0026quot; + (1/x) y' - (a^2 + p^2/x^2) y = 0\" style=\"width: 181.5px; height: 39px;\" width=\"181.5\" height=\"39\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.25px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.125px; text-align: left; transform-origin: 384px 21.125px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14.3917px 7.91667px; transform-origin: 14.3917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewith \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(x1) = y1\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 35.0083px 7.91667px; transform-origin: 35.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and either \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(x0) = y0\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.1083px 7.91667px; transform-origin: 10.1083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e or \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJQAAAAoCAYAAAAYNNPaAAADoUlEQVR4nO2bbZWzMBCFrwccYKAGUFAFdVAHdYCFaqgEPGBhNdRC3x/hHrJsSPMxSWjfec7pnyVspnND5iMUUBRFURRFURTlq+kAnCLGnwCMhWxx0S/zdRXnPDKxelXnuXz6gLEnABPqizsAmBvMe0Ri9KrOGcALwE/A2NMyrpWoF5hF9T8To1cT7jAGXt+M62C+xLm4RX7uqBtuj0aoXs14ImzXucOEutZ0MDYPrQ1pRKheTTghbLX3y7ijiDjiwFt+QUL1agaFCdmdjpS7cIG3Dr+1CdWrGTPei9LBiHcrb04UM4BHayMqE6JXNh1MKDrDiO4qJXnN7l30CNt1LogLd5013/aeDma7llic42LXp1FaryyGZeIJxrkvuCsgXttulyF9DFYV7+hhdowJJnHczney/v6CWag5xC50G/ot9xPbB6qhlxg08om/MfbiuRbyf5+R99ywOmXA2nLg7jQh3zkD0hNU276cT063upReYrDhtff0U8hYXgn3sRqh4BPkKxPOkRI+pXaoHLFL6SUKQ8rdce2ONFFTFpRty3PHHgle+OzEvIReojDf2cZehpyUMJO6oB5Yn0DfvB2M3fxMCA8lqbYdBWm9cnzphLF3m6zekH5ckSra1bJlLzSw6/2wxvC+EEd8+oKS1CvXl07Y8LOT1ZzdCUgXjU+fr6LjmG2l9sT7EpnftWUOlVtYSOqV40svP/idW4zIO0ydEwy6LHb4cgQ2TF07GCsgX0uAVV7KgjpClUck9Mr1pReu1Blr7yenGmEuFApfcRmse13nblwQrsVKwX2OZZXUsg8lUdJL6JXrSy92XJ6R30CMicPdZk47j+IW/sDaCX7nBF+o5ZjDnmsFIqFXri+92D0giYSVcX7vi45Y+0wzfj8Jti0j1uoDCHOCL9SmhOIjIqFXri/fwv6P1HvHvoNYO6dwjfnZuZ7jBOYMubvvUcjVq8qCkmyKcVt2VR7MR/bE7XeuMxymOOGKxscSwuTqlePLoH9eons8Q/YVlpxEkueC34CEXsWS8gHlXsBirJc88WYo3MJS1/Xuzw3fkTsBsnql+NILy/WSv9eS/tWJrxnncjTL6kP+fCgSab1iffkHvmA2YW0k1njn+wa5w14eF9jVDfOB7RPFlsShf+DoobReMb50Yh++lt6Ztlwgt6h6rAeZI8z3cr3p+cDnLiagjl4hvtyFr9O2Kp1rVlhd5flK0FovRVEURVEURVEURVGUEP4BkX7JQnEwa3oAAAAASUVORK5CYII=\" alt=\"y'(x0) = y'0\" style=\"width: 74px; height: 20px;\" width=\"74\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 38.1167px 7.91667px; transform-origin: 38.1167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Along with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.7px 7.91667px; transform-origin: 7.7px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003ey1\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 25.275px 7.91667px; transform-origin: 25.275px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, one of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.7px 7.91667px; transform-origin: 7.7px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003ey0\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.55px 7.91667px; transform-origin: 11.55px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eyp0\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 109.708px 7.91667px; transform-origin: 109.708px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e will be assigned numerical values, and the other will be \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.55px 7.91667px; transform-origin: 11.55px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eNaN\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 128.742px 7.91667px; transform-origin: 128.742px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The function should return the values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ey\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 80.9px 7.91667px; transform-origin: 80.9px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at the specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 194.483px 7.91667px; transform-origin: 194.483px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eOne of the applications for this equation is in groundwater. For \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"a = 1\" style=\"width: 36.5px; height: 18px;\" width=\"36.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p = 0\" style=\"width: 38px; height: 18px;\" width=\"38\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 136.567px 7.91667px; transform-origin: 136.567px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e the equation arises in the flow of water to a well pumping in a leaky confined aquifer; the independent variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 141.975px 7.91667px; transform-origin: 141.975px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the normalized distance from the well, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ey\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 8.94167px 7.91667px; transform-origin: 8.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is related to the piezometric head, a combination of the elevation and pressure of the water. Specifying the derivative at a point amounts to specifying the flow to the well.\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = solveODEB(x,coeff,Xbc,bc)\r\n% y = values of the solution\r\n% x = values of the independent variable where the solution is requested\r\n% coeff = [a p], the two parameters in the ODE\r\n% Xbc = values of x where the boundary conditions are specifified\r\n% bc = [y0 yp0 y1] = boundary values (see description)\r\n\r\n y = f(x,coeff,Xbc,bc);\r\nend","test_suite":"%%\r\ncoeff = [1 0];\r\nXbc = [1 4];\r\nbc = [1 NaN 0];\r\nx = linspace(1,4,7);\r\ny = solveODEB(x,coeff,Xbc,bc);\r\ny_correct = [1 0.505461057363874 0.265959532078956 0.140787844664873 0.071276733472728 0.029333752731051 0];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%%\r\ncoeff = [3 1];\r\nXbc = [0.5 4.5];\r\nbc = [0 NaN 2];\r\nx = [cos(pi/4) cos(pi/6) sqrt(2) (1+sqrt(5))/2 sqrt(3) sqrt(5) exp(1) pi 4+psi(1)];\r\ny = solveODEB(x,coeff,Xbc,bc);\r\ny_correct = [0.000035290165611 0.000065476121971 0.000315436935125 0.000552779648598 0.000757412623201 0.003086031722519 0.012031119481478 0.040129255417251 0.089700339919646];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%%\r\ncoeff = [1 1/2];\r\nXbc = [1 5];\r\nbc = [4 NaN -1];\r\nx = 1.2:0.75:4.95;\r\ny = solveODEB(x,coeff,Xbc,bc);\r\ny_correct = [2.974028994378906 1.041176175714286 0.308462205553512 -0.076175488441917 -0.426089583908447 -0.952692417292867];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%%\r\ncoeff = [2 4];\r\nXbc = [1 Inf];\r\nbc = [3 NaN 0];\r\nx = 1:2:9;\r\ny = solveODEB(x,coeff,Xbc,bc);\r\ny_correct = [3 0.005688558853080 0.000051725255081 0.000000653418086 0.000000009396692];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n%-------------------------------\r\n%%\r\ncoeff = [1 0];\r\nXbc = [1 4];\r\nbc = [NaN 1 0];\r\nx = linspace(1,4,7);\r\ny = solveODEB(x,coeff,Xbc,bc);\r\ny_correct = [-0.696760997897786 -0.352185550727323 -0.185310228971762 -0.098095479140575 -0.049662847941353 -0.020438614824974 0];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n \r\n%%\r\ncoeff = [3 1];\r\nXbc = [0.5 4.5];\r\nbc = [NaN 0 2];\r\nx = [cos(pi/4) cos(pi/6) sqrt(2) (1+sqrt(5))/2 sqrt(3) sqrt(5) exp(1) pi 4+psi(1)];\r\ny = solveODEB(x,coeff,Xbc,bc);\r\ny_correct = [0.000053997354167 0.000075728700374 0.000316920386259 0.000553525214653 0.000757922298088 0.003086129240342 0.012031140116004 0.040129260776539 0.089700342119009];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%%\r\ncoeff = [1 1/2];\r\nXbc = [1 5];\r\nbc = [NaN 4 -1];\r\nx = 1.2:0.75:4.95;\r\ny = solveODEB(x,coeff,Xbc,bc);\r\ny_correct = [-2.047695617799804 -0.816404083826666 -0.431398160565887 -0.374418157507686 -0.532801281305956 -0.958228725044217];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%%\r\ncoeff = [1 0];\r\nXbc = [1 Inf];\r\nbc = [NaN 3 0];\r\nx = 1:2:9;\r\ny = solveODEB(x,coeff,Xbc,bc);\r\ny_correct = [-2.098451806781317 -0.173147136186896 -0.018397012773047 -0.002117248575316 -0.000253600440751];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":2,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":8,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-05-16T01:10:22.000Z","updated_at":"2025-05-09T06:39:10.000Z","published_at":"2021-05-16T01:19:31.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve the following ordinary differential equation: \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y\u0026quot; + (1/x) y' - (a^2 + p^2/x^2) y = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\frac{d^2y}{dx^2} +\\\\frac{1}{x} \\\\frac{dy}{dx}-\\\\left(a^2+\\\\frac{p^2}{x^2}\\\\right)y = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewith \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(x1) = y1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(x_1) = y_1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and either \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(x0) = y0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(x_0) = y_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e or \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y'(x0) = y'0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\\\\prime(x_0) = y\\\\prime_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Along with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey1\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, one of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eyp0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e will be assigned numerical values, and the other will be \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eNaN\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. The function should return the values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at the specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOne of the applications for this equation is in groundwater. For \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a = 1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea = 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e the equation arises in the flow of water to a well pumping in a leaky confined aquifer; the independent variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the normalized distance from the well, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is related to the piezometric head, a combination of the elevation and pressure of the water. Specifying the derivative at a point amounts to specifying the flow to the well.\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":46025,"title":"Evaluate the gamma function","description":"The gamma function is a generalization of the factorial, and it appears in many applications such as evaluating certain integrals, working with probability distributions, and evaluating fractional derivatives. MATLAB includes the function gamma, but it accepts only real arguments.\r\nWrite a function that evaluates the gamma function for complex arguments.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 93.3333px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407.5px 46.6667px; transform-origin: 407.5px 46.6667px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63.3333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 31.6667px; text-align: left; transform-origin: 384.5px 31.6667px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12.0583px 7.66667px; transform-origin: 12.0583px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.66667px; transform-origin: 1.94167px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://mathworld.wolfram.com/GammaFunction.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003egamma function\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 302.642px 7.66667px; transform-origin: 302.642px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a generalization of the factorial, and it appears in many applications such as evaluating certain integrals, working with probability distributions, and evaluating fractional derivatives. MATLAB includes the function\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.66667px; transform-origin: 1.94167px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 19.25px 7.66667px; transform-origin: 19.25px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 19.25px 8px; transform-origin: 19.25px 8px; \"\u003egamma\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.66667px; transform-origin: 3.88333px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, but it accepts only real arguments.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 232.467px 7.66667px; transform-origin: 232.467px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that evaluates the gamma function for complex arguments.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = gamma2(z)\r\n y = f(z);\r\nend","test_suite":"%%\r\nz = 3+2i;\r\ny = gamma2(z);\r\ny_correct = -0.4226372863112003 + 0.871814255696503i;\r\nassert(abs((real(y)-real(y_correct))/real(y_correct)) \u003c 1e-6)\r\nassert(abs((imag(y)-imag(y_correct))/imag(y_correct)) \u003c 1e-6)\r\n\r\n%%\r\nz = 1+i;\r\ny = gamma2(z);\r\ny_correct = 0.4980156681183556 -0.1549498283018104i;\r\nassert(abs((real(y)-real(y_correct))/real(y_correct)) \u003c 1e-6)\r\nassert(abs((imag(y)-imag(y_correct))/imag(y_correct)) \u003c 1e-6)\r\n\r\n%%\r\nz = (1+i)/2;\r\ny = gamma2(z);\r\ny_correct = 0.818163995 - 0.7633138287i;\r\nassert(abs((real(y)-real(y_correct))/real(y_correct)) \u003c 1e-6)\r\nassert(abs((imag(y)-imag(y_correct))/imag(y_correct)) \u003c 1e-6)\r\n\r\n%%\r\nz = i;\r\ny = gamma2(z);\r\ny_correct = -0.154949828301810 - 0.4980156681183566i;\r\nassert(abs((real(y)-real(y_correct))/real(y_correct)) \u003c 1e-6)\r\nassert(abs((imag(y)-imag(y_correct))/imag(y_correct)) \u003c 1e-6)\r\n\r\n%%\r\nz = 5i;\r\ny = gamma2(z);\r\ny_correct = -0.00027170388350615125 + 0.0003399328988721375i;\r\nassert(abs((real(y)-real(y_correct))/real(y_correct)) \u003c 1e-6)\r\nassert(abs((imag(y)-imag(y_correct))/imag(y_correct)) \u003c 1e-6)\r\n\r\n%%\r\nz = 1/2 + 14.1i;\r\ny = gamma2(z);\r\ny_correct = -2.0555298837259187e-10 - 5.667644214210669e-10i;\r\nassert(abs((real(y)-real(y_correct))/real(y_correct)) \u003c 1e-6)\r\nassert(abs((imag(y)-imag(y_correct))/imag(y_correct)) \u003c 1e-6)\r\n\r\n%%\r\nz = -1+i;\r\ny = gamma2(z);\r\ny_correct = -0.1715329199082727 + 0.3264827482100833i;\r\nassert(abs((real(y)-real(y_correct))/real(y_correct)) \u003c 1e-6)\r\nassert(abs((imag(y)-imag(y_correct))/imag(y_correct)) \u003c 1e-6)\r\n\r\n%%\r\nz = -2-3i;\r\ny = gamma2(z);\r\ny_correct = -0.0001631724182726072 - 0.001128495917017955i;\r\nassert(abs((real(y)-real(y_correct))/real(y_correct)) \u003c 1e-6)\r\nassert(abs((imag(y)-imag(y_correct))/imag(y_correct)) \u003c 1e-6)\r\n\r\n%%\r\nz = 10*(rand+0.02);\r\ny_correct = gamma(z);\r\nassert(abs(gamma2(z)-y_correct)/y_correct \u003c 1e-6)\r\n\r\n%%\r\nfiletext = fileread('gamma2.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'regexp') || contains(filetext, 'system'); \r\nassert(~illegal)\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":9,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":20,"test_suite_updated_at":"2022-01-30T20:31:21.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2020-07-04T16:14:47.000Z","updated_at":"2026-01-09T11:39:54.000Z","published_at":"2020-07-05T04:43:57.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://mathworld.wolfram.com/GammaFunction.html\\\"\u003e\u003cw:r\u003e\u003cw:t\u003egamma function\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e is a generalization of the factorial, and it appears in many applications such as evaluating certain integrals, working with probability distributions, and evaluating fractional derivatives. MATLAB includes the function\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003egamma\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, but it accepts only real arguments.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that evaluates the gamma function for complex arguments.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":59541,"title":"Compute the polylogarithm for real arguments","description":"The polylogarithm appears in quantum statistics and quantum electrodynamics, and for special cases of and , it connects to the logarithm, ratios of polynomials, the zeta function, the Dirichlet eta function, and other functions. \r\nWrite a function to compute the polylogarithm for real arguments. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 74px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 37px; transform-origin: 407px 37px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 43px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.5px; text-align: left; transform-origin: 384px 21.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 57.575px 8px; transform-origin: 57.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe polylogarithm \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAE4AAAAoCAYAAABQB8xaAAAELklEQVRoQ+2aS6gPURzH792TYkVCWFDEwquUhUSJhPJOVh4LyYJiYUldNrLwyEJWHlnYKBYspDxWSrFA2bDyKnu+n5pfnc6dc+bMmfnf6X/N1Lf//c9/zsw5n/M7v8eZOzrSH1kERrNa9Y1GenCZRtCD+4/BrdTY50t3MxksULsVddsPu8UB7Yq0WfqRCY5mY9JcaXfqPYYZHNBuS2sbQjNWZrFJ8Hxw13WXddKiEvIfdO63tCZ1VnQdy+C5NFXaKT2p0TZ26XT9+KXle/K8l9It6VpVP0MWd0YNzzuN79UxY6cdS+BU8R1om6o6lPg7AwRcknUk3pPLsOKn0nLpU6xdbKn+dRqu0t9vanTALjWL4/tJKdeBu4/eqC+PpYVVg8voL03oI/4uurIGDS6z79Fmg7I2e6hNTNRYhg0cS+m1tKcl6w3N0Ff9gG8OuoJhA0fwOizNkJqkH1UrgeVKikNQKz0mClzTJNU6/774Y3HVyBv+bsExuFwHCY7AsEs6Ls2U2oiqBKxXVY67gEYAmRYBSGoVivLm547pmtLUZFDggHZEWibRCY6m4My/XdS9TldYFM//WHEN0EJ5pYELPmtQ4Nw+W1rTFFzlYJyHstQOFnLTKBLnFxIVx4UKsPQ7mL9OVnAk3vclP/dk+f6SUhLnqFuYrODKjImIPE/aK6VE5B6cQB2VTkhbpGgpVRBnSX+Xbkj46nHHMFmcOfyU4OAOlKDyUNpWsnRDbq7Snw4CHA91o1VbwYFB1klHuD53F8XABSuUtsHxwAeSm3G3CY46dY40K2Qq3nkS5pQI6t/OdnWCGwltgysrwNsEZxl9SslF2UQELfVRFeCJvktjE9QmODpKpeBn2yFwliSzcYoVse30WWIrfLVU5pjNzwUz+gIIgNlT47qUCOpzpM9RXxoCRxS66tyNMudcYJY26PwBibLqj7ResvzJzeDZQS6rMenktwLcDn1ektgB4VzZksSqOUL7ZeafuIYE9qyUEklteDb26H6fD45G2yUrkwKsgqcNDsAoi7YWQK0Bv7s+x8ooJoZjf/FJuRSqSWMbmbaDO8XroVusM8ZDxbPKgDIxb6XoEu/6ZY35LCzV3knYjGMpobKIZJY62Lc6KoKfhYUxcJYqELHeJRLLFjAhV0D7mxI7wNEl3jU4G4RbE5qvjO3AWs15WQOMvVjhOlwOvpcUiaqBydgnuS4FA62VunQNDv/m+0V2XznMQoqv4z4Y6DspJbE114GV8pIHP+rXsclvuOhJl+DMV7m7JhZMsEAGxxF7SQS8RxJ7fjkvk2w2sPJnUuVrQWvQJThLMl1fZj6Pc6QTZZbhmx7w+BeG3He2TBb3qAW+S3Bk9bMLQBbdcM53JKIvRXkuDB9u69+7BNf6YCbyhj24TNo9uB5cJoHMZr3F9eAyCWQ26y0uE9w/mWQQOMc8jDwAAAAASUVORK5CYII=\" width=\"39\" height=\"20\" alt=\"Li_n(z)\" style=\"width: 39px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 267.225px 8px; transform-origin: 267.225px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e appears in quantum statistics and quantum electrodynamics, and for special cases of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ez\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 9.325px 8px; transform-origin: 9.325px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, it connects to the logarithm, ratios of polynomials, the \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/45988\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003ezeta\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/45997\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003efunction\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 146.242px 8px; transform-origin: 146.242px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, the Dirichlet eta function, and other functions. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 11px; text-align: left; transform-origin: 384px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 142.233px 8px; transform-origin: 142.233px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to compute the polylogarithm \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"39\" height=\"20\" alt=\"Li_n(z)\" style=\"width: 39px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 62.6167px 8px; transform-origin: 62.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e for real arguments. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = polylogarithm(n,z)\r\n y = sum(z^k/k^n);\r\nend","test_suite":"%%\r\nn = 2:2:16;\r\nz = 1;\r\ny = arrayfun(@(m) polylogarithm(m,z),n);\r\ny_correct = pi.^n.*[1/6 1/90 1/945 1/9450 1/93555 691/638512875 2/18243225 3617/325641566250];\r\nassert(all(abs(y-y_correct)\u003c1e-12))\r\n\r\n%%\r\nn = 3;\r\nz = 1;\r\ny = polylogarithm(n,z);\r\ny_correct = 1.202056903159594;\r\nassert(abs(y-y_correct)\u003c1e-12)\r\n\r\n%%\r\nn = 0;\r\nz = 1./(10:-1:2);\r\ny = arrayfun(@(s) polylogarithm(n,s),z);\r\ny_correct = 1./(9:-1:1);\r\nassert(all(abs(y-y_correct)\u003c1e-12))\r\n\r\n%%\r\nn = 1;\r\nz = [-2 -3/2 -1 -1/2 0 1/10 1/7 1/5 1/3 1/2];\r\ny = arrayfun(@(s) polylogarithm(n,s),z);\r\ny_correct = [-1.098612288668110 -0.916290731874155 -0.693147180559945 -0.405465108108164 0 0.105360515657826 0.154150679827258 0.223143551314210 0.405465108108164 0.693147180559945];\r\nassert(all(abs(y-y_correct)\u003c1e-12))\r\n\r\n%%\r\nn = -1;\r\nz = [-2 -3/2 -1 -1/2 1/7 1/5 1/2];\r\ny = arrayfun(@(s) polylogarithm(n,s),z);\r\ny_correct = [-2/9 -6/25 -1/4 -2/9 7/36 5/16 2];\r\nassert(all(abs(y-y_correct)\u003c1e-12))\r\n\r\n%%\r\nn = -2;\r\nz = [-2 -3/2 -1 -1/2 0 1/17 1/13 1/11 1/5];\r\ny = arrayfun(@(s) polylogarithm(n,s),z);\r\ny_correct = [2/27 6/125 0 -2/27 0 153/2048 91/864 33/250 15/32];\r\nassert(all(abs(y-y_correct)\u003c1e-12))\r\n\r\n%%\r\nn = 1:3;\r\nz = 1/2;\r\ny = arrayfun(@(m) polylogarithm(m,z),n);\r\nzeta3 = 1.202056903159594;\r\ny_correct = [log(2) (pi^2-6*log(2)^2)/12 (4*log(2)^3-2*pi^2*log(2)+21*zeta3)/24];\r\nassert(all(abs(y-y_correct)\u003c1e-12))\r\n\r\n%%\r\nn = 2;\r\nz = -1;\r\ny = polylogarithm(n,z);\r\ny_correct = -pi^2/12;\r\nassert(abs(y-y_correct)\u003c1e-12)\r\n\r\n%%\r\nn = -1/pi;\r\nz = exp(-1);\r\ny = polylogarithm(n,z);\r\ny_correct = 0.6541056465726233;\r\nassert(abs(y-y_correct)\u003c1e-12)\r\n\r\n%%\r\nn = 4;\r\nz = 1/2;\r\ny = polylogarithm(n,z);\r\ny_correct = 0.5174790616738994;\r\nassert(abs(y-y_correct)\u003c1e-12)\r\n\r\n%%\r\nn = 1/2;\r\nz = 1/2;\r\ny = polylogarithm(n,z);\r\ny_correct = 0.8061267230428522;\r\nassert(abs(y-y_correct)\u003c1e-12)\r\n\r\n%%\r\nn = -3/2;\r\nz = -1./2.^(1:4);\r\ny = arrayfun(@(s) polylogarithm(n,s),z);\r\ny_correct = [-0.1516441361365191 -0.1313580923579523 -0.0892942267818043 -0.05260782861980105];\r\nassert(all(abs(y-y_correct)\u003c1e-12))\r\n\r\n%%\r\nn = 5*randn(1,randi(10));\r\nz = 0;\r\ny = arrayfun(@(m) polylogarithm(m,z),n);\r\ny_correct = zeros(size(n));\r\nassert(all(abs(y-y_correct)\u003c1e-12))\r\n\r\n%%\r\nfiletext = fileread('polylogarithm.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert') || contains(filetext,'regexp'); \r\nassert(~illegal)","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2024-01-07T03:51:00.000Z","deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2024-01-06T17:30:34.000Z","updated_at":"2025-02-27T16:14:20.000Z","published_at":"2024-01-06T17:31:22.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe polylogarithm \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Li_n(z)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e{\\\\rm Li}_n(z)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e appears in quantum statistics and quantum electrodynamics, and for special cases of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ez\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, it connects to the logarithm, ratios of polynomials, the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/45988\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ezeta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/45997\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003efunction\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, the Dirichlet eta function, and other functions. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to compute the polylogarithm \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Li_n(z)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e{\\\\rm Li}_n(z)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for real arguments. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":45979,"title":"Compute the perimeter of an ellipse","description":"While the area of an ellipse is straightforward to compute, the perimeter (or circumference) is more complicated. The perimeter can be expressed in terms of elliptic integrals, and several approximate formulas are available as well. \r\n\r\nCompute the perimeter of an ellipse given the lengths of the semi-major and semi-minor axes--in any order.","description_html":"\u003cp\u003eWhile the area of an ellipse is straightforward to compute, the perimeter (or circumference) is more complicated. The perimeter can be expressed in terms of elliptic integrals, and several approximate formulas are available as well.\u003c/p\u003e\u003cp\u003eCompute the perimeter of an ellipse given the lengths of the semi-major and semi-minor axes--in any order.\u003c/p\u003e","function_template":"function P = ellipsePerim(a,b)\r\n y = f(a,b);\r\nend","test_suite":"%%\r\na = 3; \r\nb = 4;\r\nP_correct = 22.103492160709504;\r\nassert(abs(ellipsePerim(a,b)-P_correct)/P_correct\u003c1e-8)\r\n\r\n%%\r\na = 4; \r\nb = 3;\r\nP_correct = 22.103492160709504;\r\nassert(abs(ellipsePerim(a,b)-P_correct)/P_correct\u003c1e-8)\r\n\r\n%%\r\na = 1;\r\nb = 8;\r\nP_correct = 32.744956600195508;\r\nassert(abs(ellipsePerim(a,b)-P_correct)/P_correct\u003c1e-8)\r\n\r\n%% \r\na = 1;\r\nb = 0.974062207869516;\r\nP_correct = 6.201967;\r\nassert(abs(ellipsePerim(a,b)-P_correct)/P_correct\u003c1e-8)\r\n\r\n%%\r\na = 4*rand(1);\r\nb = a;\r\nP_correct = 2*pi*a;\r\nassert(abs(ellipsePerim(a,b)-P_correct)/P_correct\u003c1e-8)","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":26,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-06-22T14:29:06.000Z","updated_at":"2026-01-02T12:52:12.000Z","published_at":"2020-06-22T16:17:57.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWhile the area of an ellipse is straightforward to compute, the perimeter (or circumference) is more complicated. The perimeter can be expressed in terms of elliptic integrals, and several approximate formulas are available as well.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCompute the perimeter of an ellipse given the lengths of the semi-major and semi-minor axes--in any order.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":59147,"title":"Determine aquifer properties: steady pump test in a leaky confined aquifer","description":"Problem statement\r\nWrite a function to determine the hydraulic conductivity of a leaky confined aquifer and the hydraulic conductivity of the leaky confining layer given the pumping rate and radius of the well, the drawdowns measured at distances from the well, and the thicknesses and of the leaky confined aquifer and the leaking confining layer, respectively. \r\nBackground\r\nCody Problem 59002 dealt with one-dimensional flow in a leaky confined aquifer. This problem involves flow to a well in a leaky confined aquifer. As in other pumping tests, the idea is to determine the properties of the aquifer by disturbing it, observing the response, and comparing to an analytical solution. Here the two unknown hydraulic conductivities are determined from two observations of drawdown.\r\nAn analytical solution for the drawdown can be determined by solving the equation\r\n\r\nwith the boundary conditions that the drawdown far from the well is zero (i.e., ) and the flow at the well is . Using Darcy’s law and the convention that flow to the well is positive, one finds\r\n\r\nThe governing equation is related to the one in Cody Problem 51783, and it is the polar coordinates version of the equation in Cody Problem 59002. \r\n","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 819.1px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 409.55px; transform-origin: 407px 409.55px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 64px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 32px; text-align: left; transform-origin: 384px 32px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 171.408px 8px; transform-origin: 171.408px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to determine the hydraulic conductivity \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eK\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 179.325px 8px; 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height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 9.71667px 8px; transform-origin: 9.71667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e of the leaky confining layer given the pumping rate \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"19\" height=\"20\" alt=\"Q0\" style=\"width: 19px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 36.5667px 8px; transform-origin: 36.5667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and radius \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15.5\" height=\"20\" alt=\"rw\" style=\"width: 15.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 86.35px 8px; transform-origin: 86.35px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e of the well, the drawdowns \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003es\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 74.2917px 8px; transform-origin: 74.2917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e measured at distances \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003er\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e from the well, and the thicknesses \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eb\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"18\" alt=\"b'\" style=\"width: 15px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 229.767px 8px; transform-origin: 229.767px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e of the leaky confined aquifer and the leaking confining layer, respectively. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.8333px 8px; transform-origin: 40.8333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eBackground\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/59002\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003eCody Problem 59002\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 310.817px 8px; transform-origin: 310.817px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e dealt with one-dimensional flow in a leaky confined aquifer. This problem involves flow to a well in a leaky confined aquifer. As in \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/49743\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003eother pumping tests\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 215.092px 8px; transform-origin: 215.092px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, the idea is to determine the properties of the aquifer by disturbing it, observing the response, and comparing to an analytical solution. Here the two unknown hydraulic conductivities are determined from two observations of drawdown.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 254.808px 8px; transform-origin: 254.808px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAn analytical solution for the drawdown can be determined by solving the equation\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 36.6px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.3px; text-align: left; transform-origin: 384px 18.3px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"136.5\" height=\"36.5\" alt=\"d2s/dr2 + (1/r)ds/dr - K's/Kbb' = 0\" style=\"width: 136.5px; height: 36.5px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 43px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.5px; text-align: left; transform-origin: 384px 21.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 239.6px 8px; transform-origin: 239.6px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewith the boundary conditions that the drawdown far from the well is zero (i.e., \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"60.5\" height=\"18.5\" alt=\"s(inf) = 0\" style=\"width: 60.5px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 84.4px 8px; transform-origin: 84.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e) and the flow at the well is \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACYAAAAoCAYAAACSN4jeAAAC30lEQVRYR+1XTUhVQRTWvW1qZZsWtTDUbBNJ7VwoiCIkVNCysFq2MDJciIiCLsSVP9i+BImoTRkuhMJq0SaojaAgtUoI2tf3wRw4d+7MvWce78kD78DHfe/dmTnf+c6Zc+a1tjTpaG1SXi0VsdTIVIqdeMVuQYEbwCWgQ6nxDp9fAiupCvnzU3OMhBaBdmADeAaQDEc/sOSI/sDzOnBUK0ErsdMwsAzcBP4C94AXAaOcdwC0ASR3sZHEaOyDU4J2bkdICYdVfBhzX57iOVcLOYtiu9j4qtt8Dc/7JYYY0rduzi88zzaCmPaeRroMeaOJkdMFYC+VXJFivoGyEIrtWtdluBcR0yG0qsXNHwA8KDKsDpmI+V4vYNVjYzh0+LnkjCH8ua1jirEUsDTISMmT71gkRTdFaZNiPzGLRZQjpViytPxWFlh8B4xKlxLzN7eUCNm0LvnFzUKh9PPrIeZZe58O4yesG/Ty6wq+zwD7QA/ALsH9c63LQsx6qsrKhLzXB4mFuBvI1ccQMT+UVmK6vLDBs+Hrwbw9BZxTClHBz0AuXWKnUie/JZQT2HzWsQiFkCSfA6HG/s+ty3CJEZvH5HG3IOS9VkK85m80PAT4LUj2Cx0kUToTmRgxhvMbICUjViSZN5sArzl0IJjI+F2MFxHLFPGilkQlXjlyLJQjwBcn1XkXOhZhvnsEhO5noqyFWIZ02bWHyj0BhgF9heZl8SPAazSVKrup1p2YeCw5Yj2hXEdVJddYFhj2olAmKUYD0jfZXiZVOIV06MlTOAXIQahb8ovHb7wQCgnm1WvgK7CllKEqdwFW/FFA/1GhaiwlvZ4nUi4yByyWY/R4HTgE/riN5HpdpBTf0fgdRVbmS23UNqXU5Jp9rPLzXv8+EDYS7gPY5zoBlgkO1q8dYBuInc5YS7qGNZd9R8pOZZk6qe9Jbhpg85bBf1K5/wTHTczsSEXMLJWbWClWKZaqQOr8/+6UrClQAbvIAAAAAElFTkSuQmCC\" width=\"19\" height=\"20\" alt=\"Q0\" style=\"width: 19px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Using Darcy’s law and the convention that flow to the well is positive, one finds\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 34.8px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 17.4px; text-align: left; transform-origin: 384px 17.4px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"141.5\" height=\"35\" alt=\"Q0 = -2pi rw b K (ds/dr)|_{r=rw}\" style=\"width: 141.5px; height: 35px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 146.267px 8px; transform-origin: 146.267px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe governing equation is related to the one in \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/51783\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003eCody Problem 51783\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 169.983px 8px; transform-origin: 169.983px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and it is the polar coordinates version of the equation in \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/59002\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003eCody Problem 59002\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 370.7px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 185.35px; text-align: left; transform-origin: 384px 185.35px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" width=\"504\" height=\"365\" style=\"vertical-align: baseline;width: 504px;height: 365px\" 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MeYBCJaKwqrtlBAfUY+ZBI+asWHLrtKDR77efUDV8g2bd6ntodqdrrKQNkGIBG+VE4+iD7Q0YM4A1KRVGmofgDp60yz/LS5u3PiCBuYrvn6dc9hKVsiMwqprMRB/sHLZK1Oe/WT1KtVtd3XHX94zrN+Awaprt+o9ZvuQ4BFyah977mfbT576fufuUhVv2iU0L9vH2aqZ2rJdGddM780aCd4SByX4s2SWf3l7HDl6osJ9VvJdCf11ap+bGHdR+T9yNe8orFQZiJ+ZOH7aS5ONjpfX571zbbceRsdO1XjMtiLB4yypcv7F5l2bt5dtrCSaq+1Vu8ubqwOWXbhASYK3ivPghcyCSw8dVYVfQn/ZVKB4n1n1oy9okBh/kdv28/tEYaXyQFywcceA3ldJIy0t7fdZ02WjIz9kxdK3n5/0xFPT3+mQ0NzuV1o1HrPdSPAIisohks73HY7av/fbQs+3Zjm/9OLoZk0akj0UErwl7knwwdqz//C3+w9Jpd9TUnbSSNk8+ugJddafvNk0W9Pyt21L4b//la3aLzyb9eAjmap928/7tW6TqNo+JLtLgpdGWIqTTOp37dr51vw3VdfiY7YbCR6V27Bll2xq8r7cd+jAvi1FX52OapAQ30xtYS69KJpyXiESvFUk+KCo3WVqfq3ekI0axaj5tbwh4y5t0vKSaP2eT6lSVurNC5MnSFmVxsaikrA8Cd4Fz+JjthsJHj6kostm5Os9Bwq+2HHwYEmb+MsaRjdNvqLsEmEu3NlebSR4S0jwQZESIrfe1UsF/S827/pq74kdO4r37tsjJT/24phLL47ucnUrJ55/4s9isfxg5bKhg/tKIykpacrfFjRpeqEaD4vIKfAkeJeTir656Out35zcus0jm4t2Cc1VRZc8oMHGISxI8FaR4ENOrehLyd+wdZe6goR6S19+SZ2E+Esj7ZIRVlgvllLgpcyrdlpa2v0PjAvjvvEIKfAkeLeRgF5W1LfvKfTsVm9/ieZlX1T00CHBW0KCt4NPtFLLbGUL+TuNvfqXt467suUF8m53RL23Xizlf3zK80+rHfWKpPkJz7xS82U+cgo8CV578ofY8c2B3M+2S1HfsHlXdKP6na5qedXll7S4qDEV3Q4keKtI8DVPJvier/ereq9Oeonweh9ssfzu6Impf/3zvFl/Nw9oX7wir1Vc65p8pUVOgSfBa8k83F29i1tc3Lhj29guV7diHb1mkOAtIcGHnX+9T4y7OKFVMyn2ETL9r16x9E7zkuPnvP2+GhfZc18fN2ak0YmKmvzi326/o7/RCZHIKfAkeD3Is33s5A+qoq/bWHa0TZv4y9pd3vyqhOZOXHdzNBK8VST4SGPW+zXrPQcOHlPnzIR3I3I2xXLk0IHvLVkkDfMnLHjjxWnTppkf9TZ+3MOzX3/1wUcyHxw9Tt0hJCKnwJPgHU2S+rqNO77Y+k3BFzvk/dilQ6uImny7FgneEhJ8JJOJ6ibPHnW8noR7dbTONVe2kIhfk7sBz6ZYqmvb1a7f8IvNXxtDfm7q1q5O7XPeWpoXwv+pyCnwJHjHUe87FdY3F+2RSXbProlS11tc7KTUqDESvFUk+MintuNqo1NY9O2nXxbLRkcdyJPQ6iIJE81svsSexWLZu8fV3XqkPvTo780T5Navy+v3s57SuLnP7VNmzFaDPuT/a9SQtJycnILCfTVc4GXmsfjfb234bJ3qpqSkjHzkjx06JqtuSMjDIME7hTr6Pffz7Woy3e7y5mVr6iT1iESCt4QE71BqT37el/u2bivbk9+lQ6uObWNtKvbWC7zH45FGUlKS3B4/+YOqnXXrRH2QWxTotHgp8Df3uKZxTJPX5ixq1CjaGD1rVT7mO2/tYZZ2b6GtrPIwSPCR7LujJwo9u3PW7pT3kXS7doiTos6aeoSTVz4J3hISvAZkI5X72fZ1G4tlUyVdSfad21+WGHfx2XxQ3rYthW9lv1m3bj1pq8u+njhxXLrde9yU3KWbuo+PvNzV/zt+rE/VlOw+9k8vVhKD5F8N7Nu7htfgzavqTpg0RX3SnUyYVi6e8/LzT32y/qvyu4QGCT4yyZ9b3jIbtu7aUvTVVQmtuye1TrryohDOL2E3ErwlJHjNqGT/Uf62DeXn40oi6d6pdTWuaC1zZNnq+Xxwi/h4XcAsrhw8eGDzl1+o9pXtk6vc6z78njtycnLWbigO7ea18gJ/9x035ufnX9utx+vz3jGG7EGCjxzydKmV9TXrKwjrPJkOQoK3igSvsZ27SyXWr8zd/PU3B9olNJeCHdSBQmbMNQ0bNmzMH54xOqGgDqGfMmO2BH1jKEQqL/B33HL9xg3rfU7eswMJPuzM/Vtr129qE39Zl/atel6byKnqTkeCt4QE7wZqoXHt518VfLFDup2uaimx3spC47UdLvMO8UtXe0J4zJE6Gz7kO+eVygv8pCcfnT59ujRkyvJI5lP2zXFJ8OHiKd5XsLH40y+LtxR91aljp+QrmpZ9NkR5Xed5czoSvFUkeJcwN2oq1ud+vt17wxdotd47xIc2vs98Y/6EzPtSUlKmvva2MRRSlRd42foPuLWTmrvExMTcl/5w/7t/Zcen45Dga9iGLbs+Ktj2Yd7W8xrU7dohrmeXBA6D1xIJ3hISvGtJyV+VX1RW7NdvT4hv1qV92Xqk/9bQDPFVrr5bpw6sa9uuw6y3Vti0s7TyAi++O3pi+D2/+GT1KqNf/uk4v8+aHtrHQ4KvAWonfN6X+wrWFfjshOcp0hIJ3ioSPETZ6b9eBx9J7jE/2F6F+PRRox/NeLz8vmereMf2lO7tk5KS3lyw3L7XXpUFXlm/Lm/2zBcWLFigunFxceYl9kKCBG8fn31R17Vr1rFtSxbXXYIEbwkJHt727D/8n7yt6lo6SR2vTL6iac+uCRLi312RH6r4bq5/+wjhHEJYLPCKRMDHHrxHXVJXHoM8EjWueIr33XrT1aeOHarGPgwSfMjJn0Pmo/9+/3NpWz+aBDohwVtFgkeFzGOPc9dvP7J9yUNjn3DWpTqDKvDKVQmXShX3uejegjdezMzMVO2Cwn3BXlqABB8qUteXrd6kjhJlcR0keEtI8KiSVKnHJi9W29abuiY64vqd1Sjw6miDtLS0rOdmqhHzg3A2fPpJTk4OCb7mSV1fmbtZLR7Ja+/Wnu3ZCQ/HJfhzjP/WOHmmjBYQ2P39r58y/u7fDb9F2k+/umzk43Pmv5svG1/1XWfZtqVQwrpE8++O/jdYZ899XR1LeEX7LmpE3hqLFy2UNK/O4qtdv2G98xpV4/0iBdtoWWM97mtc3eWl9cr8D+/N+Ie82OJaNJUXnrz8+v80ieoO4bhXPgkekcs/De/cXbpm/fZlH208cvTE9cmX9+p2ZaRl+koSvBSP3t3ipCE1O7lTx5PHv9v97d6dxWVXqG13dcfsf63w33yoD70lwdvNO6/fdmN7rkiDCjkuwbMGj8j1zMTxgQ5/kxC88pNCdcRTRK2MVr6LfuaMl15+/inva/iI9FGjhz/0O/+KIluTan/enTwM1uCrJHV92epN6uR16jqsYA3eEhI8QsKs9BGS6Ssv8IrUlX27ClVbfYJOoExMgrdDoLyu354JhBYJ3ioSPEIrQjK9lQJvkUrwH3yU+3HexmA/EYcE72/n7tKP8rfJK4S8jmojwVtCgkeVKtlFX4nwZvoQFnihPvKOBF9t8lCPnfxBXg8fFWwrOXiE4+FxNkjwVpHgUaWzLJbemb7GzrKzI8HnrvdwHnyw5K+/buOOZR9t2ly0p9dPrqzehxcDPkjwlpDgUaVQFUvZ1r+z8vMVa8qWvXt1b9upbax9lT6EBV6wBl8NBRt35Kzdmb9uU5cOrXpdfyXXm0OokOCtIsGjSqEtlmLn7lKJdGvWey5q2tCmz+cO4WNWCf4/q3I+2cBR9FVTh84t+8+mdgnNuye17pEUzxYGIUeCt4QEjypVbw3eCnV+1Psfr+/c4couV18Wwg8LOfsCL3Vd6pPPqXTihp69Zry+0OhUxSUJXh6JWmJXCzG9uyf0+snVLLHDJiR4q0jwiAQr12xW173v0qHVL1I7nP2u+5DvdageNyR4+dutXFMoc7We1yZyiXjUDBK8JSR4RA51QNY772/Yubv0thvbB3s4Xl7u6o9WrThx4njduvVeeDbrwUfKPiRGuj1Tb1Gnudc8jRN8oefb93M3q70vLLGjJpHgrSLBo0r27aIP5ODBA0s+3LpiTeFFTRv2LD/w3sr+3v379l7XMd7oeKnGp8CFin4J3jwnIqbRed07teYsdoQFCd4SEjyqFMbd3RIT/2/VtoJ1BRITb+zaplPblsY3AvD/sPmBg4f88YlJ4ZrFapPg5Xft+ObAP5evz12/vddPrmRXPMKIBG8VCR5VCmOBN6mFXnUudSXVxXPmg2RM1biAfAhpkODNyK72pvTsmmB8AwgfZyX4sH1cLNUdjiB15U8P3Db9fwc1i7ng6VeXPTwhW0q+1B7j22dI4R82bJjRKY/vYd+BHGx1lwRvtKoiOcZo2aNg447nXlsx7Pez9pQcHjuklzz/VHdEArtf+SFHgkfkqvk1+CpJUv/n8vXrNu7wvwiu90p8eOO7cGKC947svbpfeX3S5cY3gIjBGrwlrMHDuaQU5X62fd67eT7H4smMZNpLkyXKj/nDM+qe4eKsNXh5PtWlBhPjLr7/rus5eg6RiTV4q0jw0ECh59t33t8ggb7ntYkSOhvUPiUhPowHz5uckuCltM9alPth3tbbbmzPx8Ag8pHgLSHBo0oRuIu+Qt6B/vJmp/v+vHfYC1XkJ3i12JG7fvugn3XmnDc4AgneKhI8qiRVKnIO6rZCnVy3dZsn7B83HskJfsOWXfMX53+779Avf9aFhXY4CwneEhI8quS4Aq9IoJcCtma9p2uHuP63JIWlzEdmgpfUPvOfH0tDnhauQAfHcVyCD9tpcvJMGS1AF89MHC+3UtHvvfO6v4zte379uqMenzPzrY/379ur7lCTrJ/2pliP+9Wo7jLpkefhuX+skNL+pwduo7rDiRy315nz4BG50keNNloOMe2lyUarvMz3/2nSzIm/ahZzwe9ffO+511ZIfjW+VyMi5zz4D/O3Pvb0QpnuPDeuH6UdzuW4XMoaPBAylawprFyzed67eYlxF/8itUPLS6LtfvFHyBq8zGmefnUZJ79BG1zJzhKqO1ylZ9eEKePvvrFrGyl4z81e+53ftfBCLrwJXu2Tl//ZsUN6PXzPTVR3aMBxCZ41eEQutaTtIFWuKXRq21LKfPIVTR97euG/V35ujNojjGvwK9dsHvX4nLgWTeV/ls+GgTYcl0s5ih6Ry6FH0VthHmn/8D0piXEXGaOhE66j6OX/65V5H0qDffLQj7zyOYreEhI83EwdaT92SK/X3vr4uddWSF0M+Tui5hN8oefbx55eeGPXNuyTh5Ycl+BZgwdCJtg1hbjYpn9+5Ocd28ZKXfy/FZ8aoyFSw2vw/175+axFuTJlqfKz8wGHclwuJcEjcjn6NDnrenZN+MvYvqWHT/7u2X+F8FS6GkvwZbvl53+4p+TwH4b3YcUdGiPBW0WCR5UccSH6kFB77O+58zqplCvXbDZGz07NJHip7n95ddmlzaLl8fOmht5I8FaR4AEfiXEXjRt+y7qNxWpV3hitrhpI8J7ifY89vfDeX1x3W8/2agTQGAneKib7qJJ+p8lVSaL8w/fc1LFt7ISpi89yd719CV7Zubt05j8//svYvuyWh0uQ4K0iwaNK1VvSDqNQrSn07Jpwf//rX5n/YaHnW2MoePYleCHVXR6ePEiOlod7kOCtIsHDtfbv2/vBymXbthSqrrRfmDxh5oyXvD+TRmLxuOG3zFqU+2H+VmMoSPYleFXdHxvSq8XFTjonGDhLJHirSPDQj8U1hb88OWbo4L7jHr1f3gXD77lD2i88mzXh8Ywn//iYcY9yEo7/MLzPso82Ve+wO5sS/J79h1V2r18nbFsPICxI8FaR4FElvU+Tq1Pv/N+PGZqTk2P0o6Lq1/G9rKS8Tf70wG3rNhZXo8bbkeAlu780632V3XkLw21I8FaR4FElXU+TO3ayltx+snrVggULYmJi5i79onDHoaWrPaMerfj/9+F7bqpGjQ95gjfX3cnucCcSvFVM/+FajaKjjVZUVHZ2trr0W1xs09iWrdSgv2rU+NAmeO91d968cCcSvFUkeFRJ19PkDh44oBoTJk1pmdhVtasUbI0PYYJX16q79xfXkd3hZiR4qwgBqJKup8mZa+39BgxWDYuCqvEhTPB/eXXZbTe2j4ttqt62zM7hTiR4q9hGwLXUGnz1WK/xoUrwM9/6+JorYrtc/d/lA2bncCcSvFVsI6Afi2sK3mvw1WCxxp9lgldTcPkt+w5H3dnrGu9BwJ1I8FaxpUCVdD1NzlyDr7YHf9lj2YcbPcX7KnkfnWWClym4/PyVawofHtjZGGJeDncjwVvFlgJV0vU0Of/z3Ssk9bV964aJLRt6X+FOkbfPuOG3PP3qsmMnfzCG/Jxlgv/u6An5+aMG3chbFVBI8FaR4OFaVtbgF7zxYu9ucSdORtWtE1W7Th1j1Ev5x9KkTJi6ONDnzp1Ngpe3519eXTZ2SK9mTS4whgDXI8FbRSxAlXQ9Ta7KBD9+3MOZmZkPPpKZkpIiNf7UyZPGN34sMe6iW29s98q8D43+j51Ngn/hzVU9uybyMXGANxK8VSR4VEnX0+QqT/CSyBcvWjhlxuwHR4+Tbu36Deud1yjQ++X6pMsbN2rw75WfG30v1U7wK9ds/v6cBj27JqguAIUEbxUJHq5V+VH05zeo+8n6r27uc7vqnjp26PiRg5W8X+6987rcz7dv2LLL6J9RvQS/Z//hee/meR9YB0AhwVtFgod+LK4pdLn2+qSkJLOEV+LUD+fWrRMlCd7oB/DYkF4vvfm+RH/vt1X1Evyfpy5+/MHbmX8D/kjwVrEFQZV0PU3u9jv6z3n7/SkzZhv9wOrWq3fiZJQkeKMfgIT+P4/++WNPL/Q+qL4aCb5l8sBf3HwNB9YBFSLBW0WCR5V0PU3OInmPnDh+3EqCl3vGNDzvthvbz1+cbwwFn+A/23bw5JESlt6BQEjwVpHggcrJe6T2Od9bSfDq3XRbz/aenfvMxfigEvyBAwfWbP3+m42LjT4APyR4q0jwqJKup8lZd+qHc2vXb1jhefAVMhfjpR1Ugl+YeyotmbckUBkSvFUkeFQp5KfJHTx44O47bpSvbVsKpQpmPnxvYsuyS8X17nF1oMvFBCXkawqS4E8dOxToPHh/5zeoO+qXN/7l1WXSDirB39eneeMmzYwOgIqQ4K0iwaPmHTxQml+uuHj7kLt7LViwQI17PJ7jRw5G2mtSPZ6gErxo16Z5u8ubX9L2lqASvMwGgj0oD3AbErxVJHjUvFO1jOPDpzz7v1LmY2JiHnyk7IJxSUlJUkTP/jUZkjUFqevXdrgssWXDtvExOTk5kuA7t4uV7tDBfY17VKX/T5PqnBdTtONbo29NsAflAW7juFxa6/RpS597EXKbPKXUeFRO6mVod3oX79ie0r29areIvWzBovebNL1QdUNCynDhjkNGx4v83tiW//089dCq8IdflXBpz/+ZfF/K+UbfmkGDBlX4+AEoCbFOOomUBI/IFfIlbTPBi7++/I/QVvcKLXp7vsTxOTOfN/o2eOmZ8b17XL1wobHcoEjuD7a6k+CByjkuwbMGDxepffqwagwcPKRDx2TVDiHv+LvgjReltI9+4L6SkpJ659v4qS3HTtbyeDwZD90rZf6DlWWH1ynBFmzW4IHKsQZvFQkeVbLvNLkLL7zIaNlApfbMzEwp7caQnczPppMyP3RwXynz7y1ZJN1gCzYJHqgca/BWsQaPKgVa0q42cw2+7Ni68s9qCy35+Wm39aiZul6ll19+ObrST7Xxxxo8UDnW4C2huqPmmWvwJ04cV43Qim3Zau47BWlpaUb/DJlPSOG06cv/1w0bNkxug63uJHigcqzBW8UaPGqeuQZft2491fAxc8ZLieWXvlFfi96eb3zDsrjYplnPzVy62uNfd21S97zGRqv8UnoFhfvG/OEZaWuzBq+uTSR/GqMPhAlr8FaR4FGl0F/51esoen9Szme9NnVjUYlKxpKDRz9w36QnHzW+HYyaLPMnjpTK7cDBQ9ZuKH404/H6dYw3tTZr8OraRJ8WrDX65T5ZvUqq/q/69/IU7zOGAJuR4K0iwaNKob/y65kEX6Hb7+i/dNVn5tRzxJismJiYD1evqfZVbM0y373HTcaQDX4xYIik9scnPNeoUdk+efPxa3YUvXksofLKlGel6n/88cdb171rDAE2I8FbRYJHGFlZgz+/Qd3krtdv+Gzd9yePGkPVImU+uUs3o2ODzl26mandm2ZH0R87WctolWvR8jLVaHL5jaoB2I0EbxUJHlWy7zS5QGvw3uQlmrfmw6SkpHPrNDCGIpLMlSucLmuW4Bv9+JjBxyc8t3hF3toNxZ3atjSGAJuR4K0iwaNKIf80ucrX4H1s+PzTkpKS62/sI1HeGHIUzRL8wQMHjNYZrdskqlUJoGaQ4K0iwaPmxcU2nb1wqXzdN2JM5a9A+e7zz/xZGnJPNeI4NZzgPcX7Pli5bP26PKPvRcblq3jHdovv+rzc1QcP+pZznzV4O6jH6f+rravkSYAGSPBWkeARFsldunXs1KV+nXMqfwU+8ccxsqWe/OLfJL47dDJqd4JXpxTe0CVB2ovent+7W9zQwX37/ayn+al37y1ZJG25j9zKV0r39m3jY+68tce2LYXqDj6krvfucbXcf2Df3p3bxcpP9j5N0WcN3jyh0fso+hcmT1CDRv/H1Lf+9Nt0o3+G/KHld8m31OOUX333HTdaLNJVPgnQieM2BSR4RK6QnyanSGmvvLpnz3199uuvPvhI5u139JeuQyejdif4Q+VJ9/jx41LYRj9wnxoUhw+WVdzx4x4eOXSg1E41mJSUpBobPlt3y03J/mleKqXUdY/HY/Sjor755hv5sfJzVNdnDV79dtH43OA+FVcesNEqJw9eirH8rpiYmGu79VCPMz8/f/bMF9QdKlf5kwDNOG5TQIJH5Ar5aXJWSHUfN2ZkSkqKHdeyrUk1tgavCpvMh2YvXDplxuy77vmNdHfvLKpdv6FM0T5eV1S449Cct9+X28kv/q38X0T9adwj3luAgwcPTHg8Q7XHjZ9YULhP/pU05CfITEuN+6/BK6XfB/exAj57AjIeKXvwN/TslfPJltfnvaMep/zqK9p3UXewyPtJmPjXmf9z3wNqHDohwVtFgkcEystdLdW9bbsOL736o09fdaKaWYNXF96fMGmKzIeSu3S7uc/tffuWXdtnwuRXP9v4lUzRvD+T9/Y7+svMSRpmslfM0yWkOt47dNT5DepGRzeWxqJ3V9ato75jaQ3eytmP3j+nYOOOEyfLGr8aMsL7UEr51fJldCzweRLkGVD7fqAZErxVJHhUyb7T5CrkKd43csiAuLi4WW+t0OD1WWMJ/raf9+s3YLDROUPqeoXPYbtrrlUN72PZVEyXZ968YID6t63bJP5rqbEWfvqc+qrhw/viRVbOfvRO8PGNjX+b+/F/VKPa0tLS/J8EaIYEbxUJHlUK+Wlylfv7lCclink8nk6JTdUBWeqrhucZoVIzCV48mvEnoxWM/QeNLYBZ6X9250DV8CY1XjVq/XBMNXwEdeqj8F7Lb9y8bUxMjDTklSZfZ7NRGvVoGJaTUMNI8FaR4FFtxTu2G62QenzCc+oq9D5fYTkUwDopSxVej73GEnygEisP7LujJ2bOeOmFyROGDu579x03Dr7r1jdmTlXfNZP33m93q0bzFpVdssZKgrfCZy3/lX+8pRoyjeue1Drz4XsrfDKrFOw8A05EgreKBI9qyJ77+rUdLntr/ptGH1FRu3bt7N0tbvy4hyUKe7+t7E7w5oJ3oBL7xB/HdEpsOuHxjBeezfpg5bL8/PxPVq8yPyzfrIjFxcZ0rX79iku4YiXBB7sGLzp0TM756PMbevaStjy2BQsWyJM56clHLX4AQZVPAnRCgreKBI8qmafJSd1a9PZ8Ke3jxow0KwQUVeFmv/5q53axUlP379urxu1O8JUveEsUVivrUjunzJi9eEWe2h3y4COZ6g5mRWzcuIlq7N1T2QlvgRK892ly5kOqpDz7HEUvYlu2mvH6wqWrPfLYatcvO4d++vTpvx/je7p8hays+kMbJHirSPCokto3Lqm9e1Lr0Q/cR2mvkHd2lJp6Xcd49RG3NbYG7793+uDBAxKFpZGSkiK18+Y+t5tL6SbzX5mH2e/bW1mBD5TgvU+Tq1vPmATs2XdINfz5nE9viott+uDocbnrPXFxcdL997+yg9pX7/8kQD+Oy6W1Tp+2/QKQFdpczB4tVEHC6E9vSqKuV09aOaNjgUwIBg7oJyHb6FflhckTXng2Sxo5H30uIVgNKu8tWTRyaNkRc5LdpbqrQaXCf5VYfu05qaxLV32mRkwfrFymrgrXt/8vJ0421u9FhT/HvNqMzCrUXneT+XNu+3m/Z1/6uxqskHnPl19+OfX2/1GDgVTyJEA/kkuvjGtsdJyABI8IJa8QyXZz3ynwr1IPPpKp9vfyJV9LV//36m9K23Yd5LbfXQODCvHVXoP3D6916xk7rle9/55qKJKJ/++t2art/a/Un9jj8ficH79+XZ6qtcLKGnzzS2NVY9bfX1QNxazZwnsNXh6P/4boxJlL3Vn5INpKngTohzV4q1iDR+XUKyQutmnWczOlhgUVRt1D6lOjBv+tWFLaJTS/vfhDaUvBDqpmV3sN3uf4MnlIHZO6qvbiRQslzatBaQy4tZN5MVrvtfO70p9UDSnD2XNf379vr5ReCcf9ftZTBtWZbFbW4JO7dFM72HNyciY9+ajMD4p3bJefo6q7Wl/3XoP/+5Qnuye1ljuYBy5s21L4u8dGqnaXNpUd9KcEehLEgjdeTGzZcJHX5fTzcldXOKKWVBD5/KeDEY4EDwegzAci06AjR76ThhS2Ga8vlNJu7hIP1xq8PKRGjaLVwXQlJSUjhw68KuHStvEx0pCueeCk99p5p7YtzXMRx40ZeV3H+N7d4l54NktKu8xXEq9sL+NW1uDF2N+X7TAX06dPl/lBSvf2ahf65Bf/1uGqso+E8V6Dv7TFZfKQ5A7yG6XQytctNyWrJaHs/1t5qk4zdTcr/BN8cs+yRYpFC99QXfGvt+bKrffIZ5/my21qP+N6+4hwJHirSPAIllnmu/e4yRhyPZkox1zYXKrg0lWf+aw6B1uwg50QXHpRI9Wo8ONeHhw9TmqqCt+njpWt68vDW7wib/hDv2ta/u+8dzwIKfwyQTE/k0b+oUzm5r5TIPOVhuUf+t6oyaXqW4r5231+jtx/7tIvzKdCfo6MyO+9/Y7+TZr96CeIoem/kYmFCv2mlJQU+QkdOiYb/UpV8iTIy1V+9ddff20e0r97Z5H8D27bts0cWfPxf+S3W9lVgEjguFwatoPsNnlKqfGATSSMzp6bHWyNHzRoUKHlg+ys8xTvk2pndCzYv2+v9xXsgyVbYbVtCer3rl+XF9uy1dn8Xn/qg4tkutCpbcviHdvTbusxde6HwwdcL7cyIo+ze1Lr+9IftulTE2GHhFgnHWxBggf0ZHeCty6o6i6io8/qQGVz2xLU75XIHtrqLjpff2vTRlE7Cz+WtsxaLrm0pQrramTD55+WlJR0u77qQ/kQIRyX4FmDB/RUY2vwIafN7L/lJdE/nBujFt0XzHu9+cUXnqrTrH///mqk8Msv5PbyK64uuyucwHGvTBI8oKfISfCuJVu5uwb9Oicn57ujJ3bvLOp9x69kMC6x4+49+2XkvXeyU1JSzm9Ql7TjFCR4q3hNA7ZyboLXSc/UW+R2xdK3CwoK4tuWnT3Y+fpbvy7evvXLz2Sk36D7ZYS04xQkeKt4TQO2IsFHguQu3VrEXvbG316Mjo7ukNBcRuJim3bq1GnC+NElJSVdu/VQd4MjkOCtIsEDtiLBR4gbbuqVn5/frkNnM9Vc3CJeRm7uc3uj8jMA4RSOy6Vcix7QUGLLhrNmzTI61siEIKhr0QNuI7mUa9FbQoIHbEWCB0KLNXirWIMHbMUaPBBarMFbRYIHbEWCB0KLBG8VCR6wFQkeCC0SvFUkeMBWJHggtEjwVpHgAVuR4IHQIsFbRYIHbEWCB0KLBG8VCR6wFQkeCC0SvFUkeMBWJHggtByXS8NzJbu86bWMFgAbdE6Pqt6V7NZOM7oA/CUPC6JilpaWNm4czivfhS3BA7AVCR4II6nunTp1MjphQoEH9MQaPBBGM2fO9Hg8EydONPrhQIEH9ESCB8LlyJEjTzzxhDSefvppNRIWFHhATyR4IFyef/75kpISachtGEM8BR7QEwkeCItTp055B/cwhngKPKAnEjwQFpMmTVLxXQljiKfAA3oiwQM1zye+K+EK8RR4QE8keGhp38GopblGOwJt2bJl7NixGRkZWVlZ0pXG+PHjf/Ob3+TmBnzQpaWlOeWMvjXqX8mt0a8IF7oBNNQ5PWr23Oxga/ygQYO40A0i3JiXo1aui2p4XlT241FNGxmDNSaoC93UqmWpwk6dOnXEiBHS2LZtW3x8vBqs0sSJEzMzM6VRyb8iwQN6IsEjwhXuiLrpkbLJaIWJXAb7PFZ+Tcblxogi1V1Iaa/56h5R3nnnHbmNiYm5+OKL1Yg/CjygJ9bgEeH2How6dMRo+/B8EzVuRtneeEnqfboYgyKv0Gj0vcFouFNpaemqVaukkZ6eft5556lBfxR4QE8keES4enWMhlR6H2NeNhp/HvqjpP5+eXyXqj8otbyvhcZNmhkty+bMmaMaw4YNU40KUeABPZHgEeGOnzQaF/54Z/uvnypL8GL0XVHd25UPnbGkfGd+ml7xvXT/HqNl2fz58+U2LS2t8jV7CjygJxI8IlyFCX7m4qjPtpU1enb0jemFO8p22ouBGsX36lGH3Pe7a6DqBkKBB/REgkeE80/wS3OjXnyrrHF166hJZceV/8jStWW3Uvj1PryuqKioX79+3bp1S09Pnzp1qnSNb5yhqntcXNzdd6WpkUAo8ICeSPCIcD4JXtJ51qyyhtRv/+ouencuq+76xXfvNfjMzMzWrVsvWLDg448/nj59+ogRI6Trc4p8fHx8Wlpa1lOTjH5gFHhATyR4RDifBD/oSeOg+mmPVpzRE1uWFf7kRKOrDXMNXoK7uqjtsGHDMjIykpKS1HhqauqmTZtUW0iBz87OrjK+Cwo8oCcSPCKcmeDFr58y1tcnDI2Ku6R8yDWio6NVIz8/Py4urqSkZNq0aVlZWXl5eepyeGLUqFGqERQKPKAnEjwinJngpy8yDqwblBrV2+usd5c4cOCAakhkLyoqaty4seoKyfEpKSnSqPKqtBWiwAN6IsEjwpkJvmdHozFruZHj3Sk7O9toeenfv79qLF7640v6WUCBB/REgkeEMxN8YmzZKe+KeYkbKImJxkEH1ThdngIP6IkEjwjnvQY/KNXI8Z9ti3r8tfIh16j8SnatW7dWje3bt6uGdRR4QE8keEQ4M8Gr0+QmjTAOnv+/jyL6A2FDrvJobn4eXe3atVXDOgo8oCcSPCKcmeDNC9288BujoT5pBsK80E2LFi1UwzoKPKAnEjwinJng/7sY3/K/i/EPPm80XG7NmjWqkZycrBrWUeABPZHgEeHMBH/oqNEQ5mJ84Q63LMaba/ApKSn+58JNmzZNNRISElTDOgo8oCcSPCKcGdx9Pk0u85dlHwgrXLIYb67Bezye1NRU7xqfnp4ug9LIyMjwPj/eIgo8oCcSPCKcmeDNSq80bRQ1bbTRzppVFuX1phJ8TExMUlJSfn6+NJKTk6W0y+306dPVfR566CHVCAoFHtATCR4Rzqeue0tsGXXvLWWNQ0fKrnOnN5Xgo6Oj8/Ly1HXrpMxLaZdbaUvVLykpueSS6ly/lwIP6IkEjwjXvV1ZFZev7u2NEW8P3Bk1/p6y7yY56tNl5sxbUOsM6RqtWrVkXN3BX58+fbKysn77299Ke3k5dYVauZ09N1u61dg5r9Qyz7GrSXnTy/7PAdikc3qUbBqCrfGDBg1aaxzQA6ACycOqrpjx8fFq4dzUoUOHTz/91OjUIBI8oCcSPBAWr776qtE6I/N3fzRaNYsCD+iJNXggLFJSUuLi4oxOeXy38tntdmAXPaChAW+OuuDCNkbHsnO/PzD91j8ZHQB+rOyiFzk5Oampqao9e242BR5AyIxY/sKauQ8YHXmf16r4ne4z3jHtuVf7PGJ0APixWOCFWokP1+q7Ep4CLzYXHzZaAEKt/0Mz1i14WLVVFfev8f7jUuDn/3WoagPw8f2pU1fGWT2gPT09ffr06cOGDTMvRVfzWIMHdGbWb1XL1aAINC6bMKMF4MfODebz3ObPn2/ehgsFHtCWdzoXZi0PNC6C2oQBrmJ9+jtx4sSSkhJpyK201WDNC1uBJygAtvKp4oqq5YHGjQ6Ailic/p46derpp582OlFR3u0aFrYCT1AAbFVhzVbVPdC40QFQEYu5dNKkSSq+K2EM8SR4QFs+tdys4oHGAZwln/iuhCvEk+ABnZm13KeKBxoHcDa2bNkyduzYjIyMrKws6Upj/Pjxv/nNb3Jzw/DBt2F7b2/ylFLjAZt4nyYnAlVxn3FOkwMqlxB7gdGyIOyzZxI8oDm1lZFbo39GoHEAFXLcyjJr8IDOzAzhU8sDjQPQBgke0JZZxRWzlgcaB6ATEjygJ58qrqhaHmjc6ACoiONyKQke0FOFNVtV90DjRgdARViDt4oED9jNp5abVTzQOACdkOABnZm13KeKBxoHoA0SPKA5Vcv9q3igcQAVYg3eKhI8UDNUFVd53VugcQAVYg3eKhI8UAPMjO5TywONA9AGCR7QllnFFbOWBxoHoBMSPKAnnyquqFoeaNzoAKgIa/BWkeABW1VYs1V1DzRudABUhDV4q0jwgN18arlZxQONh8Unq1fl5a42OgBChwQP6Mys5T5VPNC43aSWJ7ZsKF8yxS/YuOPaDpcNvuvWgX1739StnXEPACFCggc0p2q5fxUPNG6rU2fe+Ns92wb0vqqkpER1Txw/rhpAxGIN3ioSPFAzVBVXed1boHH7yLS+9pk3vgR3uU1JSZm9cKl8/WrISDUORCzW4K0iwQM1wMzoPrU80LitZFpvJvg9e3YPHDxk6mtvJ3fpJl/po0arcQChQoIHtGVWccWs5YHG7ead4OPi4sb+4WnVBmAHEjygJ58qrqhaHmjc6NjGO8FnTZ5+foO6qg04AmvwVpHgAVtVWLNVdQ80bnRs453gzUoPOAVr8FaR4AG7+dRys4oHGrebd4KXSs9GALAVCR7QmVnLfap4oHFb+SR4NgKArUjwgOZULfev4oHG7UOCh6M5bkpKggc0p6q4yuveAo3bhwQPR7NpSlpaWppTzuiHDgke0JmZ0X1qeaBxW5HgAX9z5sxJLVdUVGQMhQgJHtCWWcUVs5YHGrcbCR6oSSR4QE8+VVxRtTzQuNGxDQkejua4KSkJHtBThTVbVfdA40bHHlLO5atu4zjVPeeCWDYCcBabpqSNmzQzWqFGgge05VPLzSoeaNxWqpx3atuycMch+ZKG2giwKYDLle7fY7RCjQQP6Mys5T5VPNC4faSQ+7zrVZdNAWATEjygOVXL/at4oHGbSCH3edeT4PHd0RPPTBwvXwUbd8grwefFsGf/4RcmT5DvfrJ6lTEUVjUzGS0qKsrMzExNTZXb2bNnn82h9SR4QHOqiqu87i3QuH183vWqy6bAtaSc79l3aNpLk+Ur5+0ZMuLzYvjDI7984dks+e66/FxjKKxsmox6r8FLUW/duvXEiRNzcnLkdtCgQdLNza3m/z4JHtCZmdF9anmgcfv4v+XZCMBUt249n+ouwV1d++XmPrenjxqtBrVkrsH369dPiro0hg0blpGRERdnHJH605u7Vi/Hk+ABbZlVXDFreaBxW/m/5dkIuJma3jVqYLwOT5w4rhrKe0sWSXCXRrurO06ZMVsN6io6Olo18vPzpaiXlJRMmzYtKytLirrcyvi+g1FDhw5V9wkKCR7Qk08VV1QtDzRudOxBgoc3Nb07cuQ71ZUErxpi25bCkUMHSqNZs4vfWLBcDUYCm6akBw4cUI2kpCQp6o0bN1ZdITn+2m49pJGTk1NaWqoGrSPBA3qqsGar6h5o3OjYgwQPbzK9kxfAqVoXqK6Z4L87emL4r9NU+/V575zfoK5qRwK7p6TZ2dlGy8u9vxqkGouXBj3XIcED2vKp5WYVDzRuKxI8vKnpXe3Th1XXTPCPDu/v8XikMWXG7NZtEtWgmyUmGk9CNU6XJ8EDOjNruU8VDzRuHxI8vKnpnZngFfPAukczHr+5z+1qUHuVX8kuKSlJNbZv364a1pHgAc2pWu5fxQON24QED3+Nz/3WaHkdWBexh83bNCWtPJpv3rxZNcwParKOBA9oTlVxlde9BRq3CQke3tRfv/T7i1R329bNv3lgmDQi+bD5sExJDx06pBotWrRQDetI8IDOzIzuU8sDjdsnwhO8utzKd0dPGH3YyfzTm2vwu4q3nTpWVsm+OxT0seJ6W7NmjWokJyerhnUkeEBbZhVXzFoeaNxWkZzgpd6Uft/4mYnjU65tQ5mvSeYafO/evR/NeFwaHo/nkVG/VoMuYa7Bp6Sk+J8LN23aNNVISEhQDetI8ICefKq4omp5oHGjY48IT/DNzt0ltyUlJZT5GmDO7cwEX3LoRPqo0eqAsn//Kzt77utqPKLYNCU11+BlcpOamupd49PT09U5BRkZGd7nx1tUwVu9ZmwuNv6uAEKu/0Mz1i142L+Wq5FA4x3Tnqu/a+HJ42XXHqlT7/zDhw9dcEHDUN3Kj5WfqX64orrmoM9vlMHmsa1jW16m7myrunXrffj+kvz8fKNfLiYm5r70h//nvgci6lRsPcjcThVLT/G+3t3KLsj64COZD44et3/f3p9c10Htq1+8Ii/STpOTh31lXBBV1v+NVqE58xYMHNBPXm/x8fHqRSgTnc6dO69du9Z8Te7ateuSSy5Rbess/Xo7bPKURs4OOkAzqsBLw3sTU2VbCvyYu6+oXbv2qfJsbR61K92zbHvfqnFhDkrb5z5qcMdXnq27T0q2lnh36rxWtjaWLl3qU+BFWlpa+phnWl4SzcbKJsU7tqd0by8NVeCl8d6SReoydnFxce+uKIioZ96mAj916tQRI0bI/29RUZEkeHWioEmK/fLly6sR34WlX28HEjxgH7PAC7WV8d/W+I9LgZ//1+pc8to+ZtSzr6Hak6b849W/PKS6QpX2uNim3ndDqJjPqpng00eNVmvw4pmJ483z5SLqiHp52HYUeKnr06dPb9Wq1fDhw6W79N9zp/99fklJSdeuXa/p1PmW3qnVq+6CNXhAcxVWdxFoPKKYxdW+hpB2o1NlK51i4OAhS1d7sp6bKdVdfUuNww7+V7ITwx/6nfogNUnzEbUYb9OLIT4+PisrS1V30fu2AdnZ2ZLaZfDuu9KqXd0FR9EDmjOTutE/I9C4O51Tr/GwYcM+Xlf0+ITnVGmHTbx3ivhcyU6pX+ec6bNX1q7fUNrjxozctqVQjYed43IpCR7QmZnRfWp5oHGX8N/+pI8aPeYPzzRpeqHRZxtlG1Xd1dPrfSU7k9yh5SXRT/zvU6o7ZPCdnNFQPSR4QFtmFVfMWh5o3D2sbH/YRtlElXb19JpXsvP5PHj5br8BgwcOHiLtncVf/X5MuhpHUEjwgJ58qriianmgcaPjAla2P2yjbKJKu3p6mzSqnZSUdG23Htd06lL+zR/54xOTUlJS5LuHDhqfmB5ejpvzVfBWrxkcRQ/Yp9rnwUfaUfRwCan3kV8+5UHacRS9fUjwgLZULTc6XpubQOPuYWX7wzbKPv7PrePCsSOwBg/ozKzlPlU80LhLWNn+sI2yD89tzSDBA5pTtdy/igcadwMSfHg59Ll13LyEBA9oTlVxlde9BRp3AxJ8ePk8t06p946bl5DgAZ2ZGd2nlgcadwkr2x+2UfbxeW6ZS9mEBA9oy6ziilnLA427Bwk+vBya4B2HBA/oyaeKK6qWBxo3Oi5Agg8vhyZ4x835SPCAniqs2aq6Bxo3Oi5gZfvDNso+Ps+tU+ZSjpvzkeABbfnUcrOKBxp3DyvbH7ZR9vF5bplL2YQED+jMrOU+VTzQuEuQ4MPLoQnecUjwgOZULfev4oHG3YAEH14OTfCOm/OR4AHNqSqu8rq3QONuYGX7wzbKPj7PrVPmUo6b85HgAZ2ZGd2nlgcadwkr2x+2UfbxeW6ZS9mEBA9oy6ziilnLA427Bwk+vBya4B2HBA/oyaeKK6qWBxo3Oi5Agg8vhyZ4x835SPCAniqs2aq6Bxo3Oi5gZfvDNso+Ps+tU+ZSjpvzkeABbfnUcrOKBxp3DyvbH7ZR9vF5bplL2YQED+jMrOU+VTzQuEuQ4MPLoQnecUjwgOZULfev4oHG3YAEH14OTfCOm/OR4AHNqSqu8rq3QONuYGX7wzbKPj7PrVPmUo6b85HgAZ2ZGd2nlgcadwkr2x+2UfbxeW6ZS9mEBA9oy6ziilnLA427Bwk+vBya4B2HBA/oyaeKK6qWBxo3Oi5Agg8vhyZ4x835SPCAniqs2aq6Bxo3Oi5gZfvDNso+Ps+tU+ZSjpvzkeABbfnUcrOKBxp3DyvbH7ZR9vF5bplL2YQED+jMrOU+VTzQuEuQ4MPLoQnecUjwgOZULfev4oHG3YAEH14OTfCOm/OR4AHNqSqu8rq3QONuYGX7wzbKPj7PrVPmUo6b85HgAZ2ZGd2nlgcadwkr2x+2UfbxeW6ZS9mEBA9oy6ziilnLA427Bwk+vBya4B2HBA/oyaeKK6qWBxo3Oi5Agg8vhyZ4x835SPCAniqs2aq6Bxo3Oi5gZfvDNso+Ps+tU+ZSjpvzkeABbfnUcrOKBxp3DyvbH7ZR9vF5bplL2YQED+jMrOU+VTzQuEuQ4MPLoQnecUjwgOZULfev4oHG3YAEH14OTfBWHmdRUVHOGdI1Wjk5Mq7uUJNI8IDmVBVXed1boHE3sLL9YRtlH5/n1ilzKSuPc+/evalnSNdopabKuLpDTSLBAzozM7pPLQ807hJWtj9so+zj89zqNJfq0qVLSkqK0TlDRmTc6NQgEjygLbOKK2YtDzTuHiT48HJogrdoxowZRusM/5GaQYIH9ORTxRVVywONGx0XIMGHl0MTvMXHGR8f7x3ipS0jRqdmkeABPVVYs1V1DzRudFzAyvaHbZR9fJ5bp8ylrD9O78gervguSPCAtnxquVnFA427h5XtD9so+/g8t/rNpcwQn5aWFq74LkjwgM7MWu5TxQONuwQJPrwcmuCDooL7b3/7W9UNi/C8tx8d4LuHEMDZa9gg6tDRssZ/op4+fqpO+VgQ6tU++ZOosdK4sFHU3oNlIw3Pizp0pKyh04jqWm8jhNRL1P+pVreRTD3CZ+YGUTGnTp06fPhwoxMOYSvw8qcFYBPZbqqtZzVuxfHyQFXPK2V5f0uNO/c+Qrrm/b0b3l1h/gSEkM/zbHbVcx7J5PUQVIEPu/Dsopf3jDxTgltuuQ3hrWwlVVveYtKWzCG36u2mRqy0pSGbWhWnpC3MOwhz3BH3kVv1LfPO0hZqUMgdFPNu5s+Rb6knkK+z/1LPv/lXULfmEy53EN73j6gv9bClIa8oZwlngpfnS/1pueWW2+rdCp+G3ArZJElxUoPSNt9uVbbVP1GDquvDezxi7yMN8xlQ/P+5/31EoLvJk8Pt2dyafxf/P1bk834NkOCrpp4p9axxyy231bg1N45mw/yuMOuWfFdtXlVsUpG0knEzXQWifoui/om/sN9HGj6V2/+f+99HBLqb3Kqni3b12oqVP2gE8v4/cpbwJPg//do4yE7+2D5/Zka8MeKNkaDIP5HtqWyY5FY2TOZPCDSuVOMXAZVQrzfHlUZv3m8ZZyX4MO+iB2A3VbMllKstlPnWCzQOhJxZIx39GpP/i8xX2EVfFTWbq2Q3IIBQUZtUtW2Vt565hQ00DoScBq8xNUdxlvAUeFXaHb3TBnCWQNtW6jpQJYfWrLAleDYrAABHUKXdcXudw5bg65WfbKpuAQCIZGqVwVnCdpqcPFnm4h8AABHI0XE0PAVeHarAXnoAQCRTC8rqVkKps4RtDR4AAKfgKHqr1O4OAAAcwYkLymFL8OyfBwBEPtbgg+PQJwsA4DbmGjwJ3hKOnwcAOAgJ3qq9Xp8pBABAhCPBW8UaPADAQUjwVrEGDwBwEBK8VSrBswYPAIh8Dg2l4UzwAABEPoeGUtbgAQCoAgneKhI8AMBBSPBWkeABAA5CgrdKJXjHPVkAAHciwVvFlewAAE7h0FAangLPlewAAE6hlpX5PHhLWIMHADiIhFI+D94S1uABAA7CGrxVKsGzBg8AiHyswQdBPVkAAEQ+h4ZS1uABAKgCCd4qEjwAwEFI8FaR4AEADuLEBF/r9OnTRhMAAOgiPAkeAADYigIPAICGKPAAAGiIAg8AgIYo8AAAaIgCDwCAhijwAABoiAIPAICGKPAAAGiIAg8AgIYo8AAAaIgCDwCAhijwAABoiAIPAICGKPAAAGiIAg8AgIYo8AAAaIgCDwCAhijwAABoiAIPAICGKPAAAGiIAg8AgIYo8AAAaIgCDwCAhijwAABoiAIPAICGKPAAAGiIAg8AgIYo8AAAaIgCDwCAhijwAABoiAIPAICGKPAAAGiIAg8AgIYo8AAAaIgCDwCAhijwAABoiAIPAIB2oqL+H8bx+MmMXN8xAAAAAElFTkSuQmCC\" alt=\"Steady pump test in a leaky confined aquifer\" data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [K,Kp] = steadyPumpTestLeakyConfined(r,s,Q0,rw,b,bp)\r\n% K = hydraulic conductivity of aquifer; Kp = hydraulic conductivity of confining layer\r\n% Other variables are defined in the test suite\r\n \r\n K = Q0*log(r(1)/r(2))/(2*pi*b*(s(2)-s(1));\r\n Kp = K*bp/b;\r\n \r\nend","test_suite":"%%\r\nr = [30 75]; % Distances from the well (m)\r\ns = [0.4 0.25]; % Observed drawdown (m)\r\nQ0 = 1000; % Pumping rate (m3/d)\r\nrw = 0.6; % Radius of the well (m)\r\nb = 10; % Thickness of the confined aquifer (m)\r\nbp = 1.5; % Thickness of the leaky confining layer (m)\r\n[K,Kp] = steadyPumpTestLeakyConfined(r,s,Q0,rw,b,bp);\r\nK_correct = 93.54;\r\nKp_correct = 1.82e-2;\r\nassert(abs(K-K_correct)/K_correct\u003c1e-4)\r\nassert(abs(Kp-Kp_correct)/Kp_correct\u003c1e-3)\r\n\r\n%%\r\nr = [30 75]; % Distances from the well (m)\r\ns = [0.4 0.25]; % Observed drawdown (m)\r\nQ0 = 2000; % Pumping rate (m3/d)\r\nrw = 0.6; % Radius of the well (m)\r\nb = 10; % Thickness of the confined aquifer (m)\r\nbp = 1.5; % Thickness of the leaky confining layer (m)\r\n[K,Kp] = steadyPumpTestLeakyConfined(r,s,Q0,rw,b,bp);\r\nK_correct = 187.07;\r\nKp_correct = 3.64e-2;\r\nassert(abs(K-K_correct)/K_correct\u003c1e-4)\r\nassert(abs(Kp-Kp_correct)/Kp_correct\u003c1e-3)\r\n\r\n%%\r\nr = [30 75]; % Distances from the well (m)\r\ns = [0.6 0.4]; % Observed drawdown (m)\r\nQ0 = 1000; % Pumping rate (m3/d)\r\nrw = 0.6; % Radius of the well (m)\r\nb = 10; % Thickness of the confined aquifer (m)\r\nbp = 1.5; % Thickness of the leaky confining layer (m)\r\n[K,Kp] = steadyPumpTestLeakyConfined(r,s,Q0,rw,b,bp);\r\nK_correct = 71.32;\r\nKp_correct = 7.00e-3;\r\nassert(abs(K-K_correct)/K_correct\u003c1e-4)\r\nassert(abs(Kp-Kp_correct)/Kp_correct\u003c1e-3)\r\n\r\n%%\r\nr = [30 75]; % Distances from the well (m)\r\ns = [0.6 0.4]; % Observed drawdown (m)\r\nQ0 = 1000; % Pumping rate (m3/d)\r\nrw = 0.3; % Radius of the well (m)\r\nb = 10; % Thickness of the confined aquifer (m)\r\nbp = 3; % Thickness of the leaky confining layer (m)\r\n[K,Kp] = steadyPumpTestLeakyConfined(r,s,Q0,rw,b,bp);\r\nK_correct = 71.32;\r\nKp_correct = 1.40e-2;\r\nassert(abs(K-K_correct)/K_correct\u003c1e-4)\r\nassert(abs(Kp-Kp_correct)/Kp_correct\u003c1e-3)\r\n\r\n%%\r\nr = [100 240]; % Distances from the well (m)\r\ns = [4.0 2.8]; % Observed drawdown (m)\r\nQ0 = 3500; % Pumping rate (m3/d)\r\nrw = 0.3; % Radius of the well (m)\r\nb = 35; % Thickness of the confined aquifer (m)\r\nbp = 2.3; % Thickness of the leaky confining layer (m)\r\n[K,Kp] = steadyPumpTestLeakyConfined(r,s,Q0,rw,b,bp);\r\nK_correct = 11.43;\r\nKp_correct = 3.74e-4;\r\nassert(abs(K-K_correct)/K_correct\u003c1e-4)\r\nassert(abs(Kp-Kp_correct)/Kp_correct\u003c1e-3)\r\n\r\n%%\r\nr = [100 240]; % Distances from the well (m)\r\ns = [10 8]; % Observed drawdown (m)\r\nQ0 = 3500; % Pumping rate (m3/d)\r\nrw = 0.3; % Radius of the well (m)\r\nb = 35; % Thickness of the confined aquifer (m)\r\nbp = 2.3; % Thickness of the leaky confining layer (m)\r\n[K,Kp] = steadyPumpTestLeakyConfined(r,s,Q0,rw,b,bp);\r\nK_correct = 6.959;\r\nKp_correct = 1.126e-5;\r\nassert(abs(K-K_correct)/K_correct\u003c1e-4)\r\nassert(abs(Kp-Kp_correct)/Kp_correct\u003c1e-3)\r\n\r\n%%\r\nfiletext = fileread('steadyPumpTestLeakyConfined.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert') || contains(filetext,'regexp'); \r\nassert(~illegal)\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-11-04T15:23:23.000Z","updated_at":"2026-02-12T15:06:18.000Z","published_at":"2023-11-04T15:23:23.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to determine the hydraulic conductivity \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"K\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eK\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e of a leaky confined aquifer and the hydraulic conductivity \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"K'\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eK\\\\prime\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e of the leaky confining layer given the pumping rate \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Q0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eQ_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and radius \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"rw\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003er_w\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e of the well, the drawdowns \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"s\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e measured at distances \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"r\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003er\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e from the well, and the thicknesses \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"b\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eb\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"b'\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eb\\\\prime\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e of the leaky confined aquifer and the leaking confining layer, respectively. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eBackground\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/59002\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eCody Problem 59002\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e dealt with one-dimensional flow in a leaky confined aquifer. This problem involves flow to a well in a leaky confined aquifer. As in \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/49743\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eother pumping tests\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, the idea is to determine the properties of the aquifer by disturbing it, observing the response, and comparing to an analytical solution. Here the two unknown hydraulic conductivities are determined from two observations of drawdown.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAn analytical solution for the drawdown can be determined by solving the equation\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"d2s/dr2 + (1/r)ds/dr - K's/Kbb' = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\frac{d^2s}{dr^2}+\\\\frac{1}{r}\\\\frac{ds}{dr}-\\\\frac{K\\\\prime s}{\\nKbb\\\\prime} = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewith the boundary conditions that the drawdown far from the well is zero (i.e., \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"s(inf) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es(\\\\infty) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e) and the flow at the well is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Q0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eQ_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Using Darcy’s law and the convention that flow to the well is positive, one finds\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Q0 = -2pi rw b K (ds/dr)|_{r=rw}\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eQ_0 = -2\\\\pi r_wbK\\\\frac{ds}{dr}|_{r=r_w}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe governing equation is related to the one in \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/51783\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eCody Problem 51783\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, and it is the polar coordinates version of the equation in \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/59002\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eCody Problem 59002\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"365\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" 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the incomplete elliptic integrals","description":"Elliptic integrals can be used to evaluate integrals whose integrands have the form R(x,sqrt(P(x))), where R(x,y) is a rational function in x and y and P(x) is a polynomial of degree 4 or less. They appear in calculations of arclength (as in \u003chttps://www.mathworks.com/matlabcentral/cody/problems/45979-compute-the-perimeter-of-an-ellipse Cody Problem 45979\u003e) and analysis of the motion of a pendulum. \r\n\r\nMATLAB provides a function to compute the complete elliptic integrals of the first and second kinds but not the \u003chttps://en.wikipedia.org/wiki/Elliptic_integral _incomplete_ elliptic integrals\u003e of the first, second, and third kinds.\r\n\r\nWrite a function to evaluate the three incomplete elliptic integrals. Follow MATLAB's convention of using the parameter m, which is related to the modulus k through m = k^2.\r\n ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 165.05px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 82.525px; transform-origin: 407px 82.525px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63.05px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.525px; text-align: left; transform-origin: 384px 31.525px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 256.742px 7.79167px; transform-origin: 256.742px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eElliptic integrals can be used to evaluate integrals whose integrands have the form \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"R(x,sqrt(P(x)))\" style=\"width: 79px; height: 20.5px;\" width=\"79\" height=\"20.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24.8917px 7.79167px; transform-origin: 24.8917px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"R(x,y)\" style=\"width: 46px; height: 18.5px;\" width=\"46\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 39.675px 7.79167px; transform-origin: 39.675px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a rational function in \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.79167px; transform-origin: 15.5583px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ey\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.79167px; transform-origin: 15.5583px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAD8AAAAlCAYAAAD1EjEiAAACXUlEQVRoge2Y7Y2jMBRFTw90QAM0QAVUQAfpgA6mBWqghPQwLVBDWtj9Ya6wndhgY2e1EkdCoyHJs59934cNNzc3N9+hA5pCttqCtqozAnNBex3w3P4WowemE0+fYLO046IFVgovQAf8An+s54lx+gd4be9e2/8xhm2CtSTa1bA/sTu+ep81uIsT2tUWs0ApKsnhB1hKGpyJOzfgKuOT9GaMYmrTYBZ5KGVwZXdsDHzHdt4fuN3e1951MfOu0Cw0cT1t4Hv2dx61JnOSjkKLPRKOd5vYzr+ok+FjFBlzYXcqlM3tmH/hZtueeLiEaLffTLyrrd3ex2w+KaA2lbJPOyrsbD95n6lSnJXggFnwp2XTzt62EmNhqHGzy552TY9vqMFVxqcSo0qRMwktgNTUY3Zz2OzGSlrqogcNaAJ2V7ewq2LlPcn5DuTwsMaX4ykKiqn1EL958dvagbDsxBXnlbW1+Cl5Q6r1w/AUDa7kc+VzxXlw1ZXCJef9DJ7LVeeVU34Tf5dbZQC3pb3SK2vyOfjnhqMQs7mU8M60tCmTyDlqzrjOp8xD46YsGOAmmiwDFrlZd2Q/niruUzo2VaNk7BJztUtS4kxJPDqXSy0KHc1F4RDblJXMcLW7q6MLijMsxBOWmqUFs/D+kdTejHGzFQuBjsxwtQdSlr16NaSWNLRTfifpN0z+yTLUUImJDMmXuqf7xEpcRTrEhMbpDz73x8qq77XoMbtxJXme4cF37w5OU/x+zUP3hEVvcEuipFaDo0T4z2kw9br0JGvYrMZEufv1ge9djt7c3Nz8f/wFtf/1FBNp7n8AAAAASUVORK5CYII=\" alt=\"P(x)\" style=\"width: 31.5px; height: 18.5px;\" width=\"31.5\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 254.792px 7.79167px; transform-origin: 254.792px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a polynomial of degree 4 or less. They appear in calculations of arclength (as in\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.79167px; transform-origin: 1.94167px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/45979-compute-the-perimeter-of-an-ellipse\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problem 45979\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 133.808px 7.79167px; transform-origin: 133.808px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e) and analysis of the motion of a pendulum.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 342.542px 7.79167px; transform-origin: 342.542px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eMATLAB provides a function to compute the complete elliptic integrals of the first and second kinds but not the \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Elliptic_integral\"\u003e\u003cspan style=\"perspective-origin: 33.85px 7.79167px; transform-origin: 33.85px 7.79167px; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eincomplete\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"perspective-origin: 1.94167px 7.79167px; transform-origin: 1.94167px 7.79167px; \"\u003e\u003cspan style=\"\"\u003e elliptic integrals\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 110.45px 7.79167px; transform-origin: 110.45px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e of the first, second, and third kinds.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 367.883px 7.79167px; transform-origin: 367.883px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to evaluate the three incomplete elliptic integrals. Follow MATLAB's convention of using the parameter \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003em\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.79167px; transform-origin: 3.88333px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, which is related to the modulus \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ek\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 27.6167px 7.79167px; transform-origin: 27.6167px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e through \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"m = k^2\" style=\"width: 44.5px; height: 19px;\" width=\"44.5\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.79167px; transform-origin: 1.94167px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [F,E,Pi] = ellipticIntegrals(varargin)\r\n% m = 1st argument, phi = 2nd argument, n = 3rd argument (if it's specified)\r\n F = f1(m,phi);\r\n E = f2(m,phi);\r\n Pi = f3(m,phi,n);\r\nend","test_suite":"%%\r\nm = 16/25; phi = pi/7; n = 0.2;\r\nF_correct = 0.458608414805464; E_correct = 0.439360453883539; Pi_correct = 0.464765785383336; \r\n[F,E,Pi] = ellipticIntegrals(m,phi,n);\r\nerrorF = abs(F-F_correct)/F_correct;\r\nerrorE = abs(E-E_correct)/E_correct;\r\nerrorPi = abs(Pi-Pi_correct)/Pi_correct;\r\nassert(all([errorF errorE errorPi] \u003c 1e-8))\r\n\r\n%%\r\nm = 16/25; phi = pi/2; n = 0.2;\r\nF_correct = 1.995302777664729; E_correct = 1.276349943169906; Pi_correct = 2.262478943418680; \r\n[F,E,Pi] = ellipticIntegrals(m,phi,n);\r\nerrorF = abs(F-F_correct)/F_correct;\r\nerrorE = abs(E-E_correct)/E_correct;\r\nerrorPi = abs(Pi-Pi_correct)/Pi_correct;\r\nassert(all([errorF errorE errorPi] \u003c 1e-8))\r\n\r\n%%\r\nm = 1/64; phi = pi/3; n = 0.4;\r\nF_correct = 1.049610464554263; E_correct = 1.044793820490869; Pi_correct = 1.203999963286716; \r\n[F,E,Pi] = ellipticIntegrals(m,phi,n);\r\nerrorF = abs(F-F_correct)/F_correct;\r\nerrorE = abs(E-E_correct)/E_correct;\r\nerrorPi = abs(Pi-Pi_correct)/Pi_correct;\r\nassert(all([errorF errorE errorPi] \u003c 1e-8))\r\n\r\n%%\r\nm = 1/40; phi = pi/3; n = 0.35;\r\nF_correct = 1.051071572767161; E_correct = 1.043347162655471; Pi_correct = 1.182205359771783; \r\n[F,E,Pi] = ellipticIntegrals(m,phi,n);\r\nerrorF = abs(F-F_correct)/F_correct;\r\nerrorE = abs(E-E_correct)/E_correct;\r\nerrorPi = abs(Pi-Pi_correct)/Pi_correct;\r\nassert(all([errorF errorE errorPi] \u003c 1e-8))\r\n\r\n%%\r\nm = 0; phi = rand(1); n = 0.4;\r\nF_correct = phi; E_correct = phi; \r\n[F,E,~] = ellipticIntegrals(m,phi,n);\r\nerrorF = abs(F-F_correct)/F_correct;\r\nerrorE = abs(E-E_correct)/E_correct;\r\nassert(all([errorF errorE] \u003c 1e-8))","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2022-06-01T22:25:14.000Z","deleted_by":null,"deleted_at":null,"solvers_count":15,"test_suite_updated_at":"2020-12-25T06:02:12.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2020-06-22T22:58:33.000Z","updated_at":"2026-01-09T12:16:35.000Z","published_at":"2020-06-23T04:29:36.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eElliptic integrals can be used to evaluate integrals whose integrands have the form \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"R(x,sqrt(P(x)))\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eR(x,\\\\sqrt{P(x)})\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"R(x,y)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eR(x,y)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a rational function in \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"P(x)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eP(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a polynomial of degree 4 or less. They appear in calculations of arclength (as in\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/45979-compute-the-perimeter-of-an-ellipse\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem 45979\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e) and analysis of the motion of a pendulum.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eMATLAB provides a function to compute the complete elliptic integrals of the first and second kinds but not the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Elliptic_integral\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eincomplete\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e elliptic integrals\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e of the first, second, and third kinds.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to evaluate the three incomplete elliptic integrals. Follow MATLAB's convention of using the parameter \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"m\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, which is related to the modulus \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"k\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e through \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"m = k^2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003em = k^2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":45988,"title":"Evaluate the zeta function for real arguments \u003e 1","description":"The \u003chttps://en.wikipedia.org/wiki/Riemann_zeta_function Riemann zeta function\u003e is important in number theory. In particular, the \u003chttps://en.wikipedia.org/wiki/Riemann_hypothesis Riemann hypothesis\u003e, one of the seven \u003chttps://en.wikipedia.org/wiki/Millennium_Prize_Problems Millenium Prize Problems\u003e, states that the non-trivial zeros of the zeta function all have real part equal to 1/2. The truth of the Riemann hypothesis has consequences for the distribution of prime numbers. \r\n\r\nThis problem deals only with values of the zeta function for real arguments greater than 1. For a positive integer argument x, the zeta function is the sum of the reciprocals of integers raised to the power of x. Euler showed that when x is an even integer, the value of the zeta function is proportional to pi^x, and \u003chttps://www.mathworks.com/matlabcentral/cody/problems/45939-estimate-pi-from-certain-values-of-the-zeta-function Cody Problem 45939\u003e uses this fact to estimate pi. Less is known about the zeta function for odd integer arguments, but Apery provided that zeta(3), now known as \u003chttps://en.wikipedia.org/wiki/Apéry's_constant Apery's constant\u003e, is irrational. \r\n\r\nEvaluate the zeta function for real arguments greater than 1. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 207px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 103.5px; transform-origin: 407px 103.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12.0667px 7.8px; transform-origin: 12.0667px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Riemann_zeta_function\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eRiemann zeta function\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 145.733px 7.8px; transform-origin: 145.733px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is important in number theory. In particular, the\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Riemann_hypothesis\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eRiemann hypothesis\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 55.6167px 7.8px; transform-origin: 55.6167px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, one of the seven\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Millennium_Prize_Problems\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eMillenium Prize Problems\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 335.65px 7.8px; transform-origin: 335.65px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, states that the non-trivial zeros of the zeta function all have real part equal to 1/2. The truth of the Riemann hypothesis has consequences for the distribution of prime numbers.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 105px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 52.5px; text-align: left; transform-origin: 384px 52.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 378.467px 7.8px; transform-origin: 378.467px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThis problem deals only with values of the zeta function for real arguments greater than 1. For a positive integer argument \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABwAAAAkCAYAAACaJFpUAAAA60lEQVRIie2U0Q2DIBBA3w5s4AIu4ASdgA3cwA1cgRkcwR26gjOwgv3gLhKi0kT0o+UlJuIFH3B3QKVSqVR+kgawwCDvaWyQ+GVewATMwCrPFMVt9H3dWcwlVOoBA3TAIotyyUKK0LPtRGVdaUlMGwk9hXKWw4tweUIGIU8r8H5CZkR0S0Xu4RLhrTm0hLwZtjy6u2StyFoZax61cPSom2jsCH07yvgUIz+dCL3nCc2txP1oRWYP4qtIT+mSCX0SbzLx9LrzOaFOGji+SbpMXOfPPNRCysgXR1oKLbZs0ZSU3X45KJaHdlb5Ez4sHUr70Yy5uAAAAABJRU5ErkJggg==\" alt=\"x\" style=\"width: 14px; height: 18px;\" width=\"14\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 248.15px 7.8px; transform-origin: 248.15px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e the zeta function is the sum of the reciprocals of integers raised to the power of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 80.5167px 7.8px; transform-origin: 80.5167px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Euler showed that when \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 35.7833px 7.8px; transform-origin: 35.7833px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is an even integer, the value of the zeta function is proportional to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACIAAAAmCAYAAACh1knUAAABLklEQVRYhe2WXa3EIBBGjwccYAADq6AK6qAO6mAtVMNKqIdauBqwwH1gJkAvm93tpjQ34SQ80D8+5psZCp1Op9P5CAfcsrmRuWklwAITsAFBFnfAj8znVkKUSRa+A6sIvAQnQgIwXCVCCURLLsWQ8uIrWywx0V6NZ4s8SAk7HhEwknbyznCVbwwi5CbPLMQIvV0xywcC9v5bYnXMxEgYGQHw8u28rzzlLruwMvYLjaRyrGHk/YXSrokUnZcYeUGZ+dt8VlKDaoan9F8j5FuKUAtyW7RLLi2FqAV5LmgVHSrDI6gFeS7krbrZyanl6yvXtuzawImHmOZBoKwg7Y4qxHFS0lrKTuopLcgbmObPKbmSR6J2ZG+7+48zREDcvXbV2n+Dk3srDaum0+l0/hW/ktdyQqlGSvEAAAAASUVORK5CYII=\" alt=\"pi^x\" style=\"width: 17px; height: 19px;\" width=\"17\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5667px 7.8px; transform-origin: 15.5667px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/45939-estimate-pi-from-certain-values-of-the-zeta-function\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problem 45939\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 80.8833px 7.8px; transform-origin: 80.8833px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e uses this fact to estimate \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eπ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 27.6167px 7.8px; transform-origin: 27.6167px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Less is known about the zeta function for odd integer arguments, but Apery proved that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"zeta(3)\" style=\"width: 30.5px; height: 18.5px;\" width=\"30.5\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 48.2333px 7.8px; transform-origin: 48.2333px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, now known as\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Ap%C3%A9ry's_constant\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eApery's constant\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.8833px 7.8px; transform-origin: 10.8833px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, is irrational.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 186.7px 7.8px; transform-origin: 186.7px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eEvaluate the zeta function for real arguments greater than 1.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function z = zeta1(x)\r\n z = f(x);\r\nend","test_suite":"%%\r\nx = 3/2;\r\nz_correct = 2.612375348685488;\r\nassert(abs(zeta1(x)-z_correct)/z_correct \u003c 1e-8)\r\n\r\n%% \r\nx = 2;\r\nz_correct = pi^2/6;\r\nassert(abs(zeta1(x)-z_correct)/z_correct \u003c 1e-8)\r\n\r\n%%\r\nx = 3;\r\nz_correct = 1.202056903159594;\r\nassert(abs(zeta1(x)-z_correct)/z_correct \u003c 1e-8)\r\n\r\n%%\r\nx = 4;\r\nz_correct = pi^4/90;\r\nassert(abs(zeta1(x)-z_correct)/z_correct \u003c 1e-8)\r\n\r\n%%\r\nx = 5;\r\nz_correct = 1.036927755143370;\r\nassert(abs(zeta1(x)-z_correct)/z_correct \u003c 1e-8)\r\n\r\n%%\r\nB = [1/6 -1/30 1/42 -1/30 5/66 -691/2730 7/6 -3617/510 43867/798 -174611/330];\r\nn = randi(10);\r\nx = 2*n;\r\nz_correct = (-1)^(n+1)*B(n)*(2*pi)^(2*n)/(2*factorial(2*n));\r\nassert(abs(zeta1(x)-z_correct)/z_correct \u003c 1e-8)","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":17,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-06-24T20:47:20.000Z","updated_at":"2026-01-09T12:24:36.000Z","published_at":"2020-06-24T21:36:21.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Riemann_zeta_function\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eRiemann zeta function\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e is important in number theory. In particular, the\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Riemann_hypothesis\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eRiemann hypothesis\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, one of the seven\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Millennium_Prize_Problems\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eMillenium Prize Problems\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, states that the non-trivial zeros of the zeta function all have real part equal to 1/2. The truth of the Riemann hypothesis has consequences for the distribution of prime numbers.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis problem deals only with values of the zeta function for real arguments greater than 1. For a positive integer argument \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex,\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e the zeta function is the sum of the reciprocals of integers raised to the power of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Euler showed that when \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is an even integer, the value of the zeta function is proportional to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"pi^x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi^x\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/45939-estimate-pi-from-certain-values-of-the-zeta-function\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem 45939\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e uses this fact to estimate \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Less is known about the zeta function for odd integer arguments, but Apery proved that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"zeta(3)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\zeta(3)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, now known as\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Apéry's_constant\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eApery's constant\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, is irrational.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEvaluate the zeta function for real arguments greater than 1.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":57620,"title":"Compute frequency factors for the normal distribution","description":"In frequency analysis in hydrology, the streamflow corresponding to a specified exceedance probability (or return period ) can be computed as\r\n\r\nwhere and are the mean and standard deviation of the streamflow series, respectively, and is the frequency factor. \r\nWrite a function to compute the frequency factor for the normal distribution given the exceedance probability as a vector. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 135px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 67.5px; transform-origin: 407px 67.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 43px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.5px; text-align: left; transform-origin: 384px 21.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 156.242px 8px; transform-origin: 156.242px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn frequency analysis in hydrology, the streamflow \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"QT\" style=\"width: 20.5px; height: 20px;\" width=\"20.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 164.95px 8px; transform-origin: 164.95px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e corresponding to a specified exceedance probability \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ep\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.6667px 8px; transform-origin: 32.6667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e (or return period \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAG8AAAAlCAYAAABMBrdqAAAEZUlEQVR4Xu1aOYsVQRDe/QWekRiImgiC4ploqqKpooKBIHiF4gEegXiAYmTgKhiYeKGRoHiAgYrgBQqCiRoYGHngL9DvY6eWevN6Zrq7uh/71h74eMd0V3fX13V0zYyOlGtoNTA6tDMvEx8p5A3xJijk5SXvEsRfAL7mGKaQl0Or4zKXAO+BBYW8fErOJflMRdy2XAO0Wd4a46Dfc+0447y6us9Hgz/Az66GHfc/4/5x4JZRTmP3JvLE5KXjb3x5Xf1YWO0oufeo4f9s7iKTMkjaQWAfsBT4YBhnI/reB2Yn2ATB5O1Bj8vAbeBozYIe4vf6SuJZfB5T0ukibgJvgZWGxQ+y6ywMdqoiTca1ksdEhdf+nAtpsjya+nRgQ21wLvSH+m8Tvj+otXmD349rpOZcg0U2LWQx8BE4CayohFnJ+wU5Oxy6scy1r28TeX/R0mXy4g5EkKs/LfNi7okn1cK4sMPAuQTk0fvQ8mZmmGOPSJfy6ftXAa5AywyKbpQXY13dMvk/22hXmnsNqeSnIo96+zIIHYSe8+gSxbUcwffzqTQ3CeSkIE/CitXteqkjhLx6vBvIBL1WkaZRCvKY6B0CmJH7XtTrImAuMA24UnWUeMyfL4EXdYEh5Ol4x6NDKp9ONz3Hd6Ut7T7hnuVsloI8eqarioC2ZXHdm4HdAI9VvMaAG8A1gDqeUbt3Qq8xhDwGYZ6BePEIkapyoJVm4dDqCazkkQzGutDzLYshz5VeGZbWAVIP5WFfyF2L7xMWGEKeFrIXQsS8LQpnX26CnVYh6M8zlaUAbCWPidpywJXEtS1PztRsw/Mx+2sPoufVk2f4kie7SiYRursScJNdhJW82HIYs9Ot1epc3kPPq6co4kueVE44Bl1DSEDOrvVEA1jIE9cXUw7jmZpX09FLz2s72k0c4XzJ0/GOQTVr2ScRGaFiLOTFlsN0vGsKRboc2WOZvuSx3MPMh5erJBaqKN1+KmSbseUwXfRwhSJ9POurF/uQV3/CEOMa2sgd9mzTUg6TokeTy9TJTJ/R+JCnBTQNYrG8Yc82GYNoeaGhRFuVq1rF+68AWqTzaOZDns6GplpJTG+6mJhnKYfpokfP+a2alOQZTBBXA30FiC7yfB4BWaxuMvWNIY9e4zQQk33rJLAeiiS7bySOiusiTw/A9j2p6mTSvHEuTJr4DFIqGb4ehjHrDhBToNdFD0lWaCwHAD656ZxDE3nchcsAPpCtX0/xxzcg27sZRiJCupO0XQArI/WL1Rq+CtFUSYoth3EcXfRgHvEO4Hsz84BnwBOgs07bZXkhivjf2nKDbwFiXvfQSaAr3nnpspDnpSZno9hyGIXpJDCag+iO8WueEj3l7Bt75pWih+noVciL20vyjkrMYzFdEutMStqmV8iLIy+2HMZs8jogr04W8uL0H92LlnMPCH2TQJ6cuwa+iz+Dn0UWywvnkNbDd02ClR0+VHuPQl5qjQ5QXiFvgMpOPVQhL7VGByjvH3tR7yYA92EaAAAAAElFTkSuQmCC\" alt=\"T = 1/p\" style=\"width: 55.5px; height: 18.5px;\" width=\"55.5\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 67.2917px 8px; transform-origin: 67.2917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e) can be computed as\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 11px; text-align: left; transform-origin: 384px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"QT = mu + KT sigma\" style=\"width: 89.5px; height: 20px;\" width=\"89.5\" height=\"20\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 11px; text-align: left; transform-origin: 384px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 8px; transform-origin: 21.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eμ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eσ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 249.592px 8px; transform-origin: 249.592px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are the mean and standard deviation of the streamflow series, respectively, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"KT\" style=\"width: 19.5px; height: 20px;\" width=\"19.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 74.275px 8px; transform-origin: 74.275px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the frequency factor. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 373.275px 8px; transform-origin: 373.275px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to compute the frequency factor for the normal distribution given the exceedance probability as a vector. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function KT = normFreqFactor(p)\r\n KT = trapz(QT:inf,exp(-Q.^2));\r\nend","test_suite":"%%\r\np = 0.001;\r\nKT_correct = 3.090;\r\nassert(abs(normFreqFactor(p)-KT_correct)\u003c1e-3)\r\n\r\n%%\r\np = 0.002;\r\nKT_correct = 2.878;\r\nassert(abs(normFreqFactor(p)-KT_correct)\u003c1e-3)\r\n\r\n%%\r\np = 0.01;\r\nKT_correct = 2.326;\r\nassert(abs(normFreqFactor(p)-KT_correct)\u003c1e-3)\r\n\r\n%%\r\np = 0.02;\r\nKT_correct = 2.054;\r\nassert(abs(normFreqFactor(p)-KT_correct)\u003c1e-3)\r\n\r\n%%\r\np = 0.04;\r\nKT_correct = 1.751;\r\nassert(abs(normFreqFactor(p)-KT_correct)\u003c1e-3)\r\n\r\n%%\r\np = 0.1;\r\nKT_correct = 1.282;\r\nassert(abs(normFreqFactor(p)-KT_correct)\u003c1e-3)\r\n\r\n%%\r\np = 0.2;\r\nKT_correct = 0.842;\r\nassert(abs(normFreqFactor(p)-KT_correct)\u003c1e-3)\r\n\r\n%%\r\np = 0.3;\r\nKT_correct = 0.524;\r\nassert(abs(normFreqFactor(p)-KT_correct)\u003c1e-3)\r\n\r\n%%\r\np = 0.5;\r\nKT_correct = 0;\r\nassert(abs(normFreqFactor(p)-KT_correct)\u003c1e-3)\r\n\r\n%%\r\np = 0.8;\r\nKT_correct = -0.842;\r\nassert(abs(normFreqFactor(p)-KT_correct)\u003c1e-3)\r\n\r\n%%\r\nT = [5 10 20 25 50 100 250 500 1000];\r\nKT_correct = [0.842 1.282 1.645 1.751 2.054 2.326 2.652 2.878 3.090];\r\nassert(all(abs(normFreqFactor(1./T)-KT_correct)\u003c1e-3))\r\n\r\n%%\r\np = rand(1,randi(15));\r\nK1 = normFreqFactor(p);\r\nK2 = normFreqFactor(1-p);\r\nassert(all(abs(K1+K2)\u003c1e-3))\r\n\r\n%%\r\nfiletext = fileread('normFreqFactor.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert') || contains(filetext, 'regexp') || contains(filetext, 'interp'); \r\nassert(~illegal)","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2023-01-29T19:23:19.000Z","deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-01-29T19:23:11.000Z","updated_at":"2026-01-04T12:13:24.000Z","published_at":"2023-01-29T19:23:20.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn frequency analysis in hydrology, the streamflow \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"QT\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eQ_T\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e corresponding to a specified exceedance probability \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e (or return period \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"T = 1/p\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eT = 1/p\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e) can be computed as\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"QT = mu + KT sigma\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eQ_T = \\\\mu + K_T \\\\sigma\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"mu\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\mu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"sigma\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\sigma\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e are the mean and standard deviation of the streamflow series, respectively, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"KT\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eK_T\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the frequency factor. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to compute the frequency factor for the normal distribution given the exceedance probability as a vector. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51322,"title":"Solve an ODE: diffusion problem 1","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 264.25px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 132.125px; transform-origin: 407px 132.125px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28px 7.91667px; transform-origin: 28px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 295.017px 7.91667px; transform-origin: 295.017px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn the solution of a problem involving diffusion, the following ordinary differential equation arises\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f''+a eta f' = 0\" style=\"width: 91px; height: 18px;\" width=\"91\" height=\"18\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.25px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.125px; text-align: left; transform-origin: 384px 21.125px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 7.91667px; transform-origin: 21.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 298.233px 7.91667px; transform-origin: 298.233px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a positive constant and primes denote differentiation with respect to the independent variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 45.1167px 7.91667px; transform-origin: 45.1167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The problem has the conditions \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f(0) = f0\" style=\"width: 60px; height: 20px;\" width=\"60\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f(infinity) = 0\" style=\"width: 65px; height: 19px;\" width=\"65\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 194.342px 7.91667px; transform-origin: 194.342px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve this problem—that is, return values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 69.2333px 7.91667px; transform-origin: 69.2333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.8333px 7.91667px; transform-origin: 40.8333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eBackground\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 372.133px 7.91667px; transform-origin: 372.133px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe physical problem involves diffusion of a quantity from one point where the concentration is maintained at a constant value into a medium in which the concentration is zero far away. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function f = diffusion1ODE(eta,a,f0)\r\n% eta = independent variable\r\n% a = constant\r\n% f0 = value of f at eta = 0\r\n\r\n f = g(eta,a,f0)\r\nend","test_suite":"%%\r\na = 1/2; \r\nf0 = 1;\r\neta = 0:0.2:1;\r\nf_correct = [1 0.887537083981715 0.777297410789522 0.671373240540873 0.571607644953331 0.479500122186953];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 2; \r\nf0 = 1;\r\neta = 0.5;\r\nf_correct = 0.479500122186953;\r\nassert(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10)\r\n\r\n%%\r\na = 1;\r\nf0 = 0.5;\r\neta = [0.12 0.23 0.456 0.789 1.011];\r\nf_correct = [0.452241573979416 0.409045884857994 0.324194989107724 0.215056003266342 0.156008214896009];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 2;\r\nf0 = 1;\r\neta = 1:4;\r\nf_correct = [0.157299207050285 0.004677734981047 2.209049699858544e-05 1.541725790028002e-08];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 3/4;\r\nf0 = 4/3;\r\neta = 3*logspace(-2,0,3);\r\nf_correct = [1.305696910508165 1.060016229285651 0.012499691279247];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 0.01;\r\nf0 = 1;\r\neta = rand/120;\r\nf_correct = polyval(flip([1 -0.0797885 0 0.000132981 0 -1.99471e-7]),eta);\r\nf = diffusion1ODE(eta,a,f0);\r\nassert(all(abs(f-f_correct)\u003c1e-7))","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":6,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-04-08T01:02:26.000Z","updated_at":"2025-05-04T20:55:44.000Z","published_at":"2021-04-08T01:09:52.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn the solution of a problem involving diffusion, the following ordinary differential equation arises\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f''+a eta f' = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime\\\\prime + a\\\\eta f\\\\prime = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a positive constant and primes denote differentiation with respect to the independent variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The problem has the conditions \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(0) = f0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(0) = f_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(infinity) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(\\\\infty ) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve this problem—that is, return values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eBackground\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe physical problem involves diffusion of a quantity from one point where the concentration is maintained at a constant value into a medium in which the concentration is zero far away. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":54710,"title":"Compute the period of a pendulum started from a finite initial angle","description":"Cody Problem 49830 asks for the period of a pendulum swinging through a small angle. Here the pendulum started at rest from an angle that is not necessarily small. The other assumptions are similar (no friction or drag, massless rod). \r\nWrite a function that takes the initial angle and returns , where is the length of the pendulum and is the acceleration of gravity. In the limit as the initial angle approaches zero, the function should produce .","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 94.1px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 47.05px; transform-origin: 407px 47.05px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 43px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.5px; text-align: left; transform-origin: 384px 21.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/49830\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003eCody Problem 49830\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 61.45px 8px; transform-origin: 61.45px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e asks for the period \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eT\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 240.792px 8px; transform-origin: 240.792px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e of a pendulum swinging through a small angle. Here the pendulum started at rest from an angle \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"theta0\" style=\"width: 15px; height: 20px;\" width=\"15\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 306.883px 8px; transform-origin: 306.883px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e that is not necessarily small. The other assumptions are similar (no friction or drag, massless rod). \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.1px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.05px; text-align: left; transform-origin: 384px 21.05px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 168.683px 8px; transform-origin: 168.683px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that takes the initial angle and returns \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"T\\sqrt{g/L}\" style=\"width: 52.5px; height: 20.5px;\" width=\"52.5\" height=\"20.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24.8917px 8px; transform-origin: 24.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eL\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 107.358px 8px; transform-origin: 107.358px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the length of the pendulum and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eg\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 20.6083px 8px; transform-origin: 20.6083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the acceleration of gravity. In the limit as the initial angle approaches zero, the function should produce \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"2pi\" style=\"width: 19.5px; height: 18px;\" width=\"19.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function T = pendulumPeriod(theta0)\r\n T = theta0-theta0^3/3!+theta0^5/5!+higher order terms;\r\nend","test_suite":"%%\r\nth = pi/7;\r\nT_correct = 6.363207946270837;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%%\r\nth = pi/5;\r\nT_correct = 6.44181661515865;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%% Problem 1 on p. 194 of Davis (1962)\r\nth = pi/4;\r\nT_correct = 6.534345229832591;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%%\r\nth = pi/3;\r\nT_correct = 6.743001419250384;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%%\r\nth = pi/2;\r\nT_correct = 7.416298709205487;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%%\r\nth = 36*pi/37;\r\nT_correct = 18.190113206504414;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%% \r\nth = 72*pi/73;\r\nT_correct = 20.902949604823448;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%% \r\nth = 0;\r\nT_correct = 2*pi;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%%\r\nfiletext = fileread('pendulumPeriod.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert') || contains(filetext, 'switch'); \r\nassert(~illegal)\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":3,"created_by":46909,"edited_by":46909,"edited_at":"2022-06-06T01:13:14.000Z","deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2022-06-06T00:37:45.000Z","updated_at":"2026-01-09T20:11:24.000Z","published_at":"2022-06-06T01:13:14.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/49830\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eCody Problem 49830\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e asks for the period \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"T\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eT\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e of a pendulum swinging through a small angle. Here the pendulum started at rest from an angle \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"theta0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\theta_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e that is not necessarily small. The other assumptions are similar (no friction or drag, massless rod). \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that takes the initial angle and returns \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"T\\\\sqrt{g/L}\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eT\\\\sqrt{g/L}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"L\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eL\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the length of the pendulum and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"g\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eg\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the acceleration of gravity. In the limit as the initial angle approaches zero, the function should produce \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"2pi\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e2\\\\pi\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51975,"title":"Compute a sum of Ramanujan","description":"Srinivasa Ramanujan defined the following function:\r\n\r\nWrite a function to compute for various values of . See also Cody Problems 45960 and 46000.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 105px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 52.5px; transform-origin: 407px 52.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 160.675px 7.79167px; transform-origin: 160.675px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSrinivasa Ramanujan defined the following function:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 45px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22.5px; text-align: left; transform-origin: 384px 22.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"phi(a) = 1+2 Sum[1/((a k)^3 - a k),{k,1,infinity}]\" style=\"width: 162.5px; height: 45px;\" width=\"162.5\" height=\"45\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 86.9917px 7.79167px; transform-origin: 86.9917px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to compute \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"phi(a)\" style=\"width: 32.5px; height: 18.5px;\" width=\"32.5\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 66.5083px 7.79167px; transform-origin: 66.5083px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e for various values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 82.85px 7.79167px; transform-origin: 82.85px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. See also Cody Problems \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/45960\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration: underline; text-decoration-line: underline; \"\u003e45960\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.79167px; transform-origin: 15.5583px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46000-compute-the-harmonic-numbers\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration: underline; text-decoration-line: underline; \"\u003e46000\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.79167px; transform-origin: 1.94167px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = Ramanujanphi(a)\r\n y = f(a);\r\nend","test_suite":"%%\r\na = 2;\r\ny_correct = 2*log(2);\r\nassert(abs(Ramanujanphi(a)-y_correct)\u003c1e-14)\r\n\r\n%%\r\na = 3;\r\ny_correct = log(3);\r\nassert(abs(Ramanujanphi(a)-y_correct)\u003c1e-14)\r\n\r\n%%\r\na = 4;\r\ny_correct = (3/2)*log(2);\r\nassert(abs(Ramanujanphi(a)-y_correct)\u003c1e-14)\r\n\r\n%%\r\na = 5;\r\nphi = (1+sqrt(5))/2;\r\ny_correct = (sqrt(5)/5)*log(phi)+(1/2)*log(5);\r\nassert(abs(Ramanujanphi(a)-y_correct)\u003c1e-14)\r\n\r\n%%\r\na = 6;\r\ny_correct = (1/2)*log(3)+(2/3)*log(2);\r\nassert(abs(Ramanujanphi(a)-y_correct)\u003c1e-14)\r\n\r\n%%\r\na = 7;\r\ny_correct = (2/7)*(log(14)+2*cos(pi/7)*log(cos(pi/14))+2*log(cos(3*pi/14))*sin(pi/14)-2*log(sin(pi/7))*sin(3*pi/14));\r\nassert(abs(Ramanujanphi(a)-y_correct)\u003c1e-14)\r\n\r\n%%\r\na = 8;\r\ny_correct = log(2)+(sqrt(2)/8)*log((2+sqrt(2))/(2-sqrt(2)));\r\nassert(abs(Ramanujanphi(a)-y_correct)\u003c1e-14)\r\n\r\n%%\r\na = 12;\r\ny_correct = (1/2)*log(2)+(1/4)*log(3)+(sqrt(3)/6)*log((sqrt(3)+1)/(sqrt(3)-1));\r\nassert(abs(Ramanujanphi(a)-y_correct)\u003c1e-14)\r\n\r\n%%\r\na = 18;\r\nb = pi/9;\r\ny_correct = (1/9)*(log(2)+2*log(6)+log(sqrt(3)/2)+2*cos(b)*log(cos(b/2))+2*cos(2*b)*log(cos(b))-2*cos(b)*log(sin(b/2))-2*cos(2*b)*log(sin(b))+2*log(cos(2*b))*sin(b/2)-2*log(sin(2*b))*sin(b/2));\r\nassert(abs(Ramanujanphi(a)-y_correct)\u003c1e-14)\r\n\r\n%%\r\na = [11 13 17];\r\nsum_correct = 3.003409919427940;\r\nassert(abs(sum(Ramanujanphi(a))-sum_correct)\u003c1e-14)\r\n\r\n%%\r\na = [36 54 72 100];\r\ny_correct = [1.0000515628258977 1.0000152722224909 1.0000064421348023 1.000002404321212];\r\nk = randi(4);\r\nassert(abs(Ramanujanphi(a(k))-y_correct(k))\u003c1e-14)","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":17,"test_suite_updated_at":"2021-06-05T14:29:40.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-06-05T14:22:18.000Z","updated_at":"2026-01-20T21:12:40.000Z","published_at":"2021-06-05T14:23:20.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSrinivasa Ramanujan defined the following function:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"phi(a) = 1+2 Sum[1/((a k)^3 - a k),{k,1,infinity}]\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\phi(a) = 1+2\\\\sum_{k=1}^\\\\infty\\\\frac{1}{(a k)^3 – a k}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to compute \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"phi(a)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\phi(a)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for various values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. See also Cody Problems \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/45960\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e45960\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46000-compute-the-harmonic-numbers\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e46000\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":52125,"title":"Compute the sum of reciprocals of quadratics","description":"Write a function to compute the following sum:\r\n\r\nSee also Cody Problem 46000.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 105px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 52.5px; transform-origin: 407px 52.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 143.008px 7.79167px; transform-origin: 143.008px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to compute the following sum:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 45px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22.5px; text-align: left; transform-origin: 384px 22.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y = sum(1/(n^2+an+b),{n,1,infinity})\" style=\"width: 122.5px; height: 45px;\" width=\"122.5\" height=\"45\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 29.175px 7.79167px; transform-origin: 29.175px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSee also \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46000-compute-the-harmonic-numbers\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration: underline; text-decoration-line: underline; \"\u003eCody Problem 46000\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.79167px; transform-origin: 1.94167px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = sumRecipQuad(a,b)\r\n y = sum(1/(n^2+a*n+b));\r\nend","test_suite":"%%\r\na = 3;\r\nb = 2;\r\ny_correct = 1/2;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 5;\r\nb = 6;\r\ny_correct = 1/3;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 7;\r\nb = 10;\r\ny_correct = 47/180;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 8;\r\nb = 15;\r\ny_correct = 9/40;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 10;\r\nb = 21;\r\ny_correct = 319/1680;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 9;\r\nb = 14;\r\ny_correct = 153/700;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 13;\r\nb = 22;\r\ny_correct = 42131/249480;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 14;\r\nb = 33;\r\ny_correct = 32891/221760;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 12;\r\nb = 35;\r\ny_correct = 13/84;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 16;\r\nb = 55;\r\ny_correct = 20417/166320;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 15;\r\nb = 26;\r\ny_correct = 605453/3963960;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 16;\r\nb = 39;\r\ny_correct = 485333/3603600;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 18;\r\nb = 65;\r\ny_correct = 323171/2882880;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 20;\r\nb = 91;\r\ny_correct = 30233/308880;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 24;\r\nb = 143;\r\ny_correct = 25/312;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 21;\r\nb = 38;\r\ny_correct = 158899519/1319157840;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 24;\r\nb = 95;\r\ny_correct = 19622959/217273056;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 30;\r\nb = 209;\r\ny_correct = 11171129/169303680;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 36;\r\nb = 323;\r\ny_correct = 37/684;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\na = 25;\r\nb = 46;\r\ny_correct = 265842403/2498640144;\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\nc = randi(50);\r\na = 2*c+1;\r\nb = c*(c+1);\r\ny_correct = 1/(c+1);\r\nassert(abs(sumRecipQuad(a,b)-y_correct)\u003c1e-15)\r\n\r\n%%\r\nfiletext = fileread('sumRecipQuad.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'assignin'); \r\nassert(~illegal)","published":true,"deleted":false,"likes_count":2,"comments_count":2,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":12,"test_suite_updated_at":"2021-07-03T14:09:11.000Z","rescore_all_solutions":true,"group_id":1,"created_at":"2021-06-27T14:48:37.000Z","updated_at":"2026-01-09T19:12:11.000Z","published_at":"2021-06-27T14:53:53.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to compute the following sum:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y = sum(1/(n^2+an+b),{n,1,infinity})\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey = \\\\sum_{n=1}^\\\\infty \\\\frac{1}{n^2+an+b}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSee also \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46000-compute-the-harmonic-numbers\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eCody Problem 46000\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":47874,"title":"Compute an integral of an exponential function","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 125px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 62.5px; transform-origin: 407px 62.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 72.3583px 7.91667px; transform-origin: 72.3583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThis problem builds on \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/47673-area-under-standard-normal-curve\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problem 47673\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 225.583px 7.91667px; transform-origin: 225.583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, in which Mehmet OZC asks us to compute the area under the standard normal curve. Write a function to compute the following integral:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 44px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22px; text-align: left; transform-origin: 384px 22px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 131px; height: 44px;\" width=\"131\" height=\"44\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 53.8083px 7.91667px; transform-origin: 53.8083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eYou may not use \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 30.8px 7.91667px; transform-origin: 30.8px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eintegral\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.1083px 7.91667px; transform-origin: 10.1083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e or \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.4px 7.91667px; transform-origin: 15.4px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003equad\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = IntegralExpFn(a,b,p)\r\n y = f(a,b,p);\r\nend","test_suite":"%%\r\na = log(2); b = 1; p = 1;\r\ny_correct = 1/2;\r\nassert(abs(IntegralExpFn(a,b,p)-y_correct) \u003c 1e-14)\r\n\r\n%%\r\na = Inf; b = 1; p = 1;\r\ny_correct = 1;\r\nassert(abs(IntegralExpFn(a,b,p)-y_correct) \u003c 1e-14)\r\n\r\n%%\r\na = Inf; b = 1; p = 2;\r\ny_correct = sqrt(pi)/2;\r\nassert(abs(IntegralExpFn(a,b,p)-y_correct) \u003c 1e-14)\r\n\r\n%%\r\na = 3; b = 1; p = 3;\r\ny_correct = 0.8929795115691813;\r\nassert(abs(IntegralExpFn(a,b,p)-y_correct) \u003c 1e-14)\r\n\r\n%%\r\na = 1; b = 1/2; p = 1.5;\r\ny_correct = 0.8278055502117507;\r\nassert(abs(IntegralExpFn(a,b,p)-y_correct) \u003c 1e-14)\r\n\r\n%%\r\na = Inf; b = randi(10); p = 1/2;\r\ny_correct = 2/b^2;\r\nassert(abs(IntegralExpFn(a,b,p)-y_correct) \u003c 1e-14)\r\n\r\n%%\r\na = 0; b = 4; p = 7/2;\r\ny_correct = 0;\r\nassert(abs(IntegralExpFn(a,b,p)-y_correct) \u003c 1e-14)\r\n\r\n%%\r\nfiletext = fileread('IntegralExpFn.m');\r\nillegalfns = ~isempty(strfind(filetext, 'integral')) || ~isempty(strfind(filetext, 'quad')); \r\nassert(~illegalfns,'Please do not use integral or quad')","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":6,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-12-10T04:36:31.000Z","updated_at":"2026-01-09T17:40:58.000Z","published_at":"2020-12-10T05:15:17.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis problem builds on \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/47673-area-under-standard-normal-curve\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem 47673\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, in which Mehmet OZC asks us to compute the area under the standard normal curve. Write a function to compute the following integral:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey = \\\\int_0^a \\\\exp(-b x^p) dx\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYou may not use \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eintegral\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr/\u003e\u003cw:t\u003e or \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003equad\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr/\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":46081,"title":"Set Soldner's constant","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 93px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 46.5px; transform-origin: 407px 46.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://mathworld.wolfram.com/SoldnersConstantDigits.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eSoldner's constant\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eμ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 199.375px 7.91667px; transform-origin: 199.375px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e= 1.451369234883381... is connected to the logarithmic integral \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"li(x)\" style=\"width: 30.5px; height: 19px;\" width=\"30.5\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 74.2833px 7.91667px; transform-origin: 74.2833px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, which is the subject of \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46066\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problem 46066\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 358.017px 7.91667px; transform-origin: 358.017px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSet Soldner's constant. The test suite puts up minor resistance against directly entering the number or using simple arithmetic. Of course, you can thwart the tests easily, but I encourage you to learn about the definition of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eμ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function mu = SoldnersConstant\r\n mu = ...;\r\nend","test_suite":"%%\r\nmu_correct = 1.451369234883381;\r\nassert(abs(SoldnersConstant-mu_correct) \u003c 1e-14)\r\n\r\n%%\r\nfiletext = fileread('SoldnersConstant.m');\r\nassert(isempty(strfind(filetext,'45136923488')), 'Please do not set the constant directly')\r\nbanfns = ~isempty(strfind(filetext, 'sum')) || ~isempty(strfind(filetext, 'plus')) ||...\r\n ~isempty(strfind(filetext, '+')) || ~isempty(strfind(filetext, 'minus')) || ...\r\n ~isempty(strfind(filetext, '-')) || ~isempty(strfind(filetext, 'diff')) || ...\r\n ~isempty(strfind(filetext, 'str2num'));\r\nassert(~banfns, 'Please do not set the constant indirectly with arithmetic')\r\n\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":7,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":16,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-07-30T02:45:56.000Z","updated_at":"2026-02-01T10:54:44.000Z","published_at":"2020-07-30T02:56:47.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://mathworld.wolfram.com/SoldnersConstantDigits.html\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eSoldner's constant\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"mu\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\mu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e= 1.451369234883381... is connected to the logarithmic integral \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"li(x)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e{\\\\rm li}(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, which is the subject of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46066\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem 46066\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSet Soldner's constant. The test suite puts up minor resistance against directly entering the number or using simple arithmetic. Of course, you can thwart the tests easily, but I encourage you to learn about the definition of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"mu\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\mu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":52120,"title":"Compute the fractional derivative","description":"Cody Problem 1370 asks us to compute the derivative of a polynomial. This problem extends that idea to fractional derivatives, which appear in some models of mixing in rivers and other applications. Denote the th derivative as . Then a familiar example from calculus would be . \r\nFractional calculus involves derivatives in which the order is not an integer. With and , then \r\n\r\nWrite a function that computes the fractional derivative of order of an expression of the form\r\n\r\nThe first input to the function will be a 2x matrix in which the first row is the coefficients and the second row is the exponents . The output should be in a similar form. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 256.55px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 128.275px; transform-origin: 407px 128.275px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/1370\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration: underline; text-decoration-line: underline; \"\u003eCody Problem 1370\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 294.442px 7.79167px; transform-origin: 294.442px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e asks us to compute the derivative of a polynomial. This problem extends that idea to fractional derivatives, which appear in some models of mixing in rivers and other applications. Denote the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eq\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 49.0083px 7.79167px; transform-origin: 49.0083px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eth derivative as \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"D^q x^a\" style=\"width: 32px; height: 19px;\" width=\"32\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.79167px; transform-origin: 3.88333px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Then a familiar example from calculus would be \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"D^2 x^3 = 6 x\" style=\"width: 65.5px; height: 19px;\" width=\"65.5\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.79167px; transform-origin: 3.88333px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 179.725px 7.79167px; transform-origin: 179.725px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFractional calculus involves derivatives in which the order \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eq\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 71.1667px 7.79167px; transform-origin: 71.1667px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is not an integer. With \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"q = 1/2\" style=\"width: 51.5px; height: 18.5px;\" width=\"51.5\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.79167px; transform-origin: 15.5583px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"a = 2\" style=\"width: 36.5px; height: 18px;\" width=\"36.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 19.4417px 7.79167px; transform-origin: 19.4417px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, then \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 36.9167px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.4583px; text-align: left; transform-origin: 384px 18.4583px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"D^{1/2} x^2 = 8 x^{3/2} / (3 sqrt(pi))\" style=\"width: 117px; height: 37px;\" width=\"117\" height=\"37\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 196.292px 7.79167px; transform-origin: 196.292px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that computes the fractional derivative of order \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eq\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 88.675px 7.79167px; transform-origin: 88.675px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e of an expression of the form\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 26px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 13px; text-align: left; transform-origin: 384px 13px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f(x) = c­1 x^a1 + c2 x^a2 + c3 x^a3 +...\" style=\"width: 204px; height: 26px;\" width=\"204\" height=\"26\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 43.6333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.8167px; text-align: left; transform-origin: 384px 21.8167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 124.85px 7.79167px; transform-origin: 124.85px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe first input to the function will be a 2x\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 143.775px 7.79167px; transform-origin: 143.775px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e matrix in which the first row is the coefficients \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABkAAAAoCAYAAAALz1FrAAAA9ElEQVRIie2VbQ2EMAyGHw9zMAMYOAUoOAc4wAEW0IAEPGABDWdh92NtRgjf2y65ZE/CDxi06du3BQqFQqEQjQVqoJXLpgz+AkZgBt5yPwAOaFIk6CTYsHquSWbAxCToJdC4cTbL2RSToJEgH7a1t3jZHldhJLjDy5UFrcIB1c1vDd559dmLI0Gqu3K8CYY4RKt40lSt5FSBmCSXmTieAYO3d5TrWkI16wZW+J6tJ90S5sqdJVCWze8kcY+vbk/vmu3tcEhFWIZXmqkrKMk+20N7mXQ7L7H8wJE6hNnWEARn6b8mi2TqxoGMjVcn3l2ohcI/8AWjzEUn2YFV5AAAAABJRU5ErkJggg==\" alt=\"ci\" style=\"width: 12.5px; height: 20px;\" width=\"12.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 83.625px 7.79167px; transform-origin: 83.625px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and the second row is the exponents \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABsAAAAoCAYAAAAPOoFWAAABF0lEQVRYhe2UUQ3DIBCGPw84mAEMoGAK5qAO6mAW0FAJeJiFaqiF7aF3KWNlkJQt2cKX8FD+9n7uehx0Op1O52ewwACMgJM9J8+2pUkAZjFzgAduwF3WqYXRRYLNgEm0e6R91Cg2ux41OkfB3I5+KujVGGCRQFPmHc16YT/raka2U+e6LIgeMrqlsks1q1wgFx1mzLzjRffvjGoCBcqZa2ZvSxyX8LyjD2yZN/1fqdmF9RpMPJfZ81oFS0WX2sgssJ7cSDC9b5qZlxV3rKPcPE/oyeM1sZVsSfZTrqINNWawlnCUD9K5Vxq+OjObDecchobzsoROlsPzsga9zHvXpjmzmBkqu/EI+r9ufKFBtIsPTZZO5894AM2JYzwwHTn5AAAAAElFTkSuQmCC\" alt=\"ai\" style=\"width: 13.5px; height: 20px;\" width=\"13.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 124.85px 7.79167px; transform-origin: 124.85px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The output should be in a similar form. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = fractDeriv(x,q)\r\n % x = 2xn matrix with coefficients in the first row and exponents in the second row\r\n % q = order of the derivative\r\n % y = output matrix with coefficients in the first row and exponents in the second row\r\n \r\n y = q*x^(q-1);\r\nend","test_suite":"%% Example from problem description\r\nx = [1; 2];\r\nq = 1/2;\r\ny_correct = [8/(3*sqrt(pi)); 3/2];\r\nassert(all(abs(fractDeriv(x,q)-y_correct)\u003c1e-10))\r\n\r\n%% Constant\r\nc = rand;\r\nx = [c; 0];\r\nq = 1/2;\r\ny_correct = [c/sqrt(pi); -1/2];\r\nassert(all(abs(fractDeriv(x,q)-y_correct)\u003c1e-10))\r\n\r\n%% Polynomial #1\r\nx = [3 -7 4; 2 1 0];\r\nq = 1/2;\r\ny_correct = [[8 -14 4]/sqrt(pi); 3/2 1/2 -1/2];\r\nassert(all(abs(fractDeriv(x,q)-y_correct)\u003c1e-10,'all'))\r\n\r\n%% Polynomial #2\r\nx = [1:4; 3:-1:0];\r\nq = 1/3;\r\ny_correct = [1.495438426033838 2.658557201837934 3.323196502297416 2.953952446486593; 8/3 5/3 2/3 -1/3];\r\nassert(all(abs(fractDeriv(x,q)-y_correct)\u003c1e-10,'all'))\r\n\r\n%% Quadratic term\r\nx = [7; 2];\r\nq = 3/2;\r\ny_correct = [28/sqrt(pi); 1/2];\r\nassert(all(abs(fractDeriv(x,q)-y_correct)\u003c1e-10))\r\n\r\n%% Two fractional derivatives amounting to a first derivative\r\nq = rand;\r\nx = [6; 5];\r\nyy_correct = [30; 4];\r\nassert(all(abs(fractDeriv(fractDeriv(x,q),1-q)-yy_correct)\u003c1e-10))\r\n\r\n%% Two fractional derivatives undoing each other\r\nq = rand;\r\nx = [5; 2];\r\nassert(all(abs(fractDeriv(fractDeriv(x,q),-q)-x)\u003c1e-10))\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":8,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-06-27T04:55:37.000Z","updated_at":"2026-01-09T18:33:26.000Z","published_at":"2021-06-27T05:00:18.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/1370\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eCody Problem 1370\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e asks us to compute the derivative of a polynomial. This problem extends that idea to fractional derivatives, which appear in some models of mixing in rivers and other applications. Denote the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"q\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eq\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eth derivative as \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"D^q x^a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eD^q x^a\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Then a familiar example from calculus would be \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"D^2 x^3 = 6 x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eD^2 x^3 = 6 x\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFractional calculus involves derivatives in which the order \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"q\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eq\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is not an integer. With \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"q = 1/2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eq = 1/2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a = 2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea = 2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, then \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"D^{1/2} x^2 = 8 x^{3/2} / (3 sqrt(pi))\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eD^{1/2} x^2 = \\\\frac{8}{3\\\\sqrt{\\\\pi}} x^{3/2}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that computes the fractional derivative of order \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"q\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eq\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e of an expression of the form\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(x) = c­1 x^a1 + c2 x^a2 + c3 x^a3 +...\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(x) = c­_1 x^{a_1} + c_2 x^{a_2} + c_3 x^{a_3} + \\\\dots \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe first input to the function will be a 2x\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e matrix in which the first row is the coefficients \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"ci\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec_i\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and the second row is the exponents \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"ai\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea_i\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The output should be in a similar form. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51745,"title":"Solve an ODE: equation A","description":"Write a function to solve the following ordinary differential equation: \r\n\r\nwith and . The function should return the values of at the specified values of . One application of this equation involves the propagation of internal gravity waves in a fluid with variable density gradient.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 118.6px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 59.3px; transform-origin: 407px 59.3px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 209.417px 8px; transform-origin: 209.417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve the following ordinary differential equation: \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 36.6px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.3px; text-align: left; transform-origin: 384px 18.3px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y\u0026quot;-xy = 0\" style=\"width: 74px; height: 36.5px;\" width=\"74\" height=\"36.5\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 43px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.5px; text-align: left; transform-origin: 384px 21.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14.3917px 8px; transform-origin: 14.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewith \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(x0) = y0\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(x1) = y1\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 128.742px 8px; transform-origin: 128.742px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The function should return the values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ey\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 80.9px 8px; transform-origin: 80.9px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at the specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 62.2333px 8px; transform-origin: 62.2333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. One application of this equation involves the propagation of internal gravity waves in a fluid with variable density gradient.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = solveODEA(x,x0,y0,x1,y1)\r\n y(x0) = y0; \r\n y(x1) = y1; \r\n y = f(x,x0,y0,x1,y1);\r\nend","test_suite":"%%\r\nx0 = -3; \r\ny0 = 1;\r\nx1 = 3;\r\ny1 = 0.2;\r\nx = linspace(x0,x1,7);\r\ny = solveODEA(x,x0,y0,x1,y1);\r\ny_correct = [1 -0.608544735138462 -1.416513846161609 -0.930561687980622 -0.339539877942166 -0.041385541117497 0.2];\r\nassert(all(abs(y-y_correct) \u003c 1e-13))\r\n\r\n%%\r\nx0 = -5; \r\ny0 = 2;\r\nx1 = 5;\r\ny1 = 0;\r\nx = linspace(x0,x1,11);\r\ny = solveODEA(x,x0,y0,x1,y1);\r\ny_correct = [2 -0.400646543833150 -2.159956361501116 1.296651996575315 3.053707895612597 2.024329503005815 0.771421025528832 0.199130370987114 0.037568752980204 0.005346964905914 0];\r\nassert(all(abs(y-y_correct) \u003c 1e-13))\r\n\r\n%%\r\nx0 = -4; \r\ny0 = -1;\r\nx1 = 2;\r\ny1 = 0.3;\r\nx = linspace(x0,x1,6);\r\ny = solveODEA(x,x0,y0,x1,y1);\r\ny_correct = [-1 -4.088820829832713 5.996215644909088 6.297137060461841 2.304927532867409 0.3];\r\nassert(all(abs(y-y_correct) \u003c 1e-13))\r\n\r\n%%\r\nx0 = -7; \r\ny0 = 0;\r\nx1 = 3;\r\ny1 = 0.1;\r\nx = linspace(x0,x1,6);\r\ny = solveODEA(x,x0,y0,x1,y1);\r\ny_correct = [0 -0.004972738967217 0.002891447704152 -0.005345046731767 0.007070410943182 0.1];\r\nassert(all(abs(y-y_correct) \u003c 1e-14))\r\n\r\n%% anti-cheating--product of two values\r\nx0 = -2; \r\ny0 = 1;\r\nx1 = 2;\r\ny1 = 0.1;\r\nx = [-1 1];\r\nz = prod(solveODEA(x,x0,y0,x1,y1)); \r\nz_correct = 1.336786968358133;\r\nassert(all(abs(z-z_correct) \u003c 1e-13))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":3,"created_by":46909,"edited_by":46909,"edited_at":"2022-06-15T13:16:15.000Z","deleted_by":null,"deleted_at":null,"solvers_count":7,"test_suite_updated_at":"2021-05-14T12:28:46.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-05-11T04:16:02.000Z","updated_at":"2025-09-02T13:25:13.000Z","published_at":"2021-05-11T04:19:47.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve the following ordinary differential equation: \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y\u0026quot;-xy = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\frac{d^2y}{dx^2} – x y = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewith \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(x0) = y0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(x_0) = y_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(x1) = y1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(x_1) = y_1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The function should return the values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at the specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. One application of this equation involves the propagation of internal gravity waves in a fluid with variable density gradient.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":58394,"title":"Integrate a product of gamma functions","description":"Write a function to compute the following integral:\r\n\r\nwhere and is the gamma function, the subject of Cody Problem 46025.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 104.1px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 52.05px; transform-origin: 407px 52.05px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 152.742px 8px; transform-origin: 152.742px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to compute the following integral:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 44px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22px; text-align: left; transform-origin: 384px 22px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 196.5px; height: 44px;\" width=\"196.5\" height=\"44\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21.1px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.55px; text-align: left; transform-origin: 384px 10.55px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 8px; transform-origin: 21.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 56px; height: 20px;\" width=\"56\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 29.5px; height: 18.5px;\" width=\"29.5\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 117.842px 8px; transform-origin: 117.842px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the gamma function, the subject of \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46025\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problem 46025\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = intGammaProduct(a)\r\n f = @(x) gamma(1+i*a*x)*gamma(1-i*a*x);\r\n y = trapz(x,f);\r\nend","test_suite":"%%\r\na = tan(1);\r\nI_correct = 1.008596722571773;\r\nassert(abs(intGammaProduct(a)-I_correct)\u003c1e-13)\r\n\r\n%%\r\na = sqrt(2);\r\nI_correct = 1.110720734539592;\r\nassert(abs(intGammaProduct(a)-I_correct)\u003c1e-13)\r\n\r\n%%\r\na = log(3);\r\nI_correct = 1.429800433690064;\r\nassert(abs(intGammaProduct(a)-I_correct)\u003c1e-13)\r\n\r\n%%\r\na = exp(4);\r\nI_correct = 0.028770138289325;\r\nassert(abs(intGammaProduct(a)-I_correct)\u003c1e-13)\r\n\r\n%%\r\na = sinh(5);\r\nI_correct = 0.021168845856719;\r\nassert(abs(intGammaProduct(a)-I_correct)\u003c1e-13)\r\n\r\n%%\r\na = asinh(6);\r\nI_correct = 0.630391294450658;\r\nassert(abs(intGammaProduct(a)-I_correct)\u003c1e-13)","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2023-06-03T19:50:53.000Z","deleted_by":null,"deleted_at":null,"solvers_count":8,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-06-03T14:18:04.000Z","updated_at":"2026-01-26T04:29:04.000Z","published_at":"2023-06-03T14:18:04.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to compute the following integral:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eI = \\\\int_{-\\\\infty}^\\\\infty \\\\Gamma(1+iax)\\\\Gamma(1-iax) dx\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ei = \\\\sqrt{-1}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Gamma(z)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the gamma function, the subject of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46025\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem 46025\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51137,"title":"Compute an integral of a product of sinusoids","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 104px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 52px; transform-origin: 407px 52px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 150.8px 7.91667px; transform-origin: 150.8px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to compute the following integral\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 44px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22px; text-align: left; transform-origin: 384px 22px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"I = integral(sin(x)^m cos(x)^n, {x,0,pi})\" style=\"width: 139px; height: 44px;\" width=\"139\" height=\"44\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 7.91667px; transform-origin: 21.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003em\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 96.5917px 7.91667px; transform-origin: 96.5917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are integers. You may not use \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 30.8px 7.91667px; transform-origin: 30.8px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eintegral\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.1083px 7.91667px; transform-origin: 10.1083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e or \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.4px 7.91667px; transform-origin: 15.4px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003equad\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 99.1917px 7.91667px; transform-origin: 99.1917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e but other functions are allowed.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = intSinmCosn(m,n)\r\n y = f(m,n);\r\nend","test_suite":"%%\r\nm = 1;\r\nn = 0;\r\ny_correct = 2;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 0;\r\nn = 1;\r\ny_correct = 0;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 0;\r\nn = 2;\r\ny_correct = pi/2;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 1;\r\nn = 2;\r\ny_correct = 2/3;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 2;\r\nn = 2;\r\ny_correct = pi/8;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 3;\r\nn = 2;\r\ny_correct = 4/15;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 0;\r\nn = 4;\r\ny_correct = 3*pi/8;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 1;\r\nn = 4;\r\ny_correct = 2/5;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 2;\r\nn = 4;\r\ny_correct = pi/16;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 3;\r\nn = 4;\r\ny_correct = 4/35;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 4;\r\nn = 4;\r\ny_correct = 3*pi/128;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 5;\r\nn = 4;\r\ny_correct = 16/315;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 6;\r\nn = 4;\r\ny_correct = 3*pi/256;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 7;\r\nn = 4;\r\ny_correct = 32/1155;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 0;\r\nn = 6;\r\ny_correct = 5*pi/16;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 1;\r\nn = 6;\r\ny_correct = 2/7;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 2;\r\nn = 6;\r\ny_correct = 5*pi/128;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 3;\r\nn = 6;\r\ny_correct = 4/63;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 4;\r\nn = 6;\r\ny_correct = 3*pi/256;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 5;\r\nn = 6;\r\ny_correct = 16/693;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 6;\r\nn = 6;\r\ny_correct = 5*pi/1024;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 7;\r\nn = 6;\r\ny_correct = 32/3003;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 0;\r\nn = 8;\r\ny_correct = 35*pi/128;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 1;\r\nn = 8;\r\ny_correct = 2/9;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 2;\r\nn = 8;\r\ny_correct = 7*pi/256;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 3;\r\nn = 8;\r\ny_correct = 4/99;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 4;\r\nn = 8;\r\ny_correct = 7*pi/1024;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 5;\r\nn = 8;\r\ny_correct = 16/1287;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 6;\r\nn = 8;\r\ny_correct = 5*pi/2048;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 7;\r\nn = 8;\r\ny_correct = 32/6435;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 2*randi(9);\r\nn = m+1;\r\ny_correct = 0;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 1;\r\nn = 22;\r\ny_correct = 2/23;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nm = 1;\r\nn = 28;\r\ny_correct = 2/29;\r\nassert(abs(intSinmCosn(m,n)-y_correct)\u003c1e-12)\r\n\r\n%%\r\nfiletext = fileread('intSinmCosn.m');\r\nillegalfns = ~isempty(strfind(filetext, 'integral')) || ~isempty(strfind(filetext, 'quad')); \r\nassert(~illegalfns,'Please do not use integral or quad')","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":10,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-03-23T00:29:43.000Z","updated_at":"2026-02-22T14:27:23.000Z","published_at":"2021-03-23T00:33:35.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to compute the following integral\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"I = integral(sin(x)^m cos(x)^n, {x,0,pi})\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eI = \\\\int_0^\\\\pi \\\\sin^mx\\\\cos^nxdx\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"m\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e are integers. You may not use \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eintegral\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e or \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003equad\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e but other functions are allowed.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":57452,"title":"Design a well field in an infinite aquifer","description":"A well field provides water for a community. The design of a well field involves a goal to meet a specified service demand (i.e., volume of water per time) with the constraint of lowering the water table by no more than , the maximum drawdown. Inputs to the design are properties of the aquifer (the hydraulic conductivity , the specific yield , and the initial saturated thickness ) and the radius of the well. \r\nThe Gupta/Chin method for designing a well field has the following steps:\r\nCompute , an initial estimate of the pumping rate, such that the drawdown at one well (i.e., at a distance ) is . Compute the transmissivity to be . Evaluate the drawdown at a time 1 year. Realize that for small values of the unconfined well function* can be approximated and compute the pumping rate from where and . \r\nCompute the number of wells by dividing the demand by the initial estimate of the pumping rate and rounding up to the nearest integer: \r\nSet the pumping rate to . \r\nArrange the wells so that they are equidistant from the central well.\r\nDetermine the distance between the central well and others so that the total drawdown at the central well is . In other words, add the drawdown from the central well to the drawdown from the other wells. If , then \r\n \r\n Write a function to design a well field using this method.\r\n\r\n\r\n*http://www.aqtesolv.com/neuman.htm","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 895.383px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 447.692px; transform-origin: 407px 447.692px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 87px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 43.5px; text-align: left; transform-origin: 384px 43.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 375.242px 8px; transform-origin: 375.242px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eA well field provides water for a community. The design of a well field involves a goal to meet a specified service demand \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"Q_d\" style=\"width: 19.5px; height: 20px;\" width=\"19.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 292.875px 8px; transform-origin: 292.875px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e (i.e., volume of water per time) with the constraint of lowering the water table by no more than \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"s_max\" style=\"width: 25.5px; height: 20px;\" width=\"25.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 47.8333px 8px; transform-origin: 47.8333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, the maximum drawdown. Inputs to the design are properties of the aquifer (the hydraulic conductivity \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eK\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 57.175px 8px; transform-origin: 57.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, the specific yield \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"S_y\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 29.1667px 8px; transform-origin: 29.1667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and the initial saturated thickness \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eb\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 50.5667px 8px; transform-origin: 50.5667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e) and the radius \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"r_w\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 37.3333px 8px; transform-origin: 37.3333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e of the well. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 226.008px 8px; transform-origin: 226.008px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe Gupta/Chin method for designing a well field has the following steps:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003col style=\"block-size: 231.783px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: decimal; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 391px 115.892px; transform-origin: 391px 115.892px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 104.05px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 52.025px; text-align: left; transform-origin: 363px 52.025px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 30.3417px 8px; transform-origin: 30.3417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eCompute \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"Q_w\" style=\"width: 20.5px; height: 20px;\" width=\"20.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 292.492px 8px; transform-origin: 292.492px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, an initial estimate of the pumping rate, such that the drawdown at one well (i.e., at a distance \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"r = r_w\" style=\"width: 40.5px; height: 20px;\" width=\"40.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 9.56667px 8px; transform-origin: 9.56667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e) is \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"s_max/2\" style=\"width: 40.5px; height: 20px;\" width=\"40.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 107.342px 8px; transform-origin: 107.342px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Compute the transmissivity to be \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"T = K(b-s_{max}/2)\" style=\"width: 116.5px; height: 20px;\" width=\"116.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 107.35px 8px; transform-origin: 107.35px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Evaluate the drawdown at a time \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"t = \" style=\"width: 22px; height: 18px;\" width=\"22\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 50.175px 8px; transform-origin: 50.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e 1 year. Realize that for small values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"u = S_y r^2/4Tt\" style=\"width: 81.5px; height: 21px;\" width=\"81.5\" height=\"21\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 243.133px 8px; transform-origin: 243.133px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e the unconfined well function* can be approximated and compute the pumping rate from \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"s_max/2 = (Q_w/4 pi T) W(u_w)\" style=\"width: 111.5px; height: 38px;\" width=\"111.5\" height=\"38\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 22.95px 8px; transform-origin: 22.95px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAOAAAABHCAYAAAAeNEu3AAANB0lEQVR4Xu2d18s1SRHGd++8MXtlYjGAgqCYEVdQMKAgiooR+UAxI4iucUXErKggoqvsjWJGEZZVTCiYMKOgKBgQFa/M/gFaP5hnqbe/numemZ6ZPmdroPi+95yeDtX1dIWu7nPlFfEEB4IDh3HgysNajoaDA8GBKwKAIQTBgQM5EAA8kPnRdHAgABgyEBw4kAMBwAOZH00HBwKAIQPBgQM5EAA8kPnRdHAgABgyEBw4kAMBwAOZH01vwoG7Wa0PNrqr0VeNfmHEZ081+qXRlzdpdWGlAcCFjIvXuuTA7a1X9zZ6mNG7jN5u9DmjLxj9xOj3Rm/oqecBwJ5mI/rSigNoPMAG6HgeZ/T3VpW3rCcA2JKbUVcrDrx6RkVjZuXvrI67Gz3T6DMz6tu1aABwV3Z33djDrXfPMkJ7/MHojYnW4PNHG93aSL7VVgP6yoyKv2hlP5IpD+iebnQ/I/zALp8AYJfTsnunXmgtvsPonUaPMnqs0T+Nnmj0XaPHG33C6LauZ5+1/z9j957WNYgvCIgfaPSiEYDW1bRxqQDgxgw+uPr7WvuAS1rtUwOgfLfkL13tvvuQ/f/FRvhRTzH6ltHXjD4+vPjmQbgJciwNaqBxCZbwfD/TrzWse5u9zGLBGHpeKOI0xJpZ7vxdaa1HWj8xwQAjQHqOkQ/F4289zehByXh+PIAMTQj4vLZDw/zW6B9G95jBB957ntFrhzo/aP++f2hnDZh9FwD2tUYEXv5nxCLyECM0YjrGGV3fpugpaEAYihnU4mG1/7dRlxGxFgMc6hBAPmx/S0PJJ8J/Qzj1IJiUT4VTmpFyOT9KWvIOlfxkAbh+AJsPjLAAsGVAxHINQFgMABuaD8Axx37Mb20oR82mqncAMsnfHhjZYtAIAZP9GqNuHfMGA5VQe7NSn6WaBgCycX3PDJDGQEsXqQ9NlnsvHYK0L+BIfTJp2rUaEI1/F6PPu3FoA57PCCx19/QMwNbgE/OZFEwqfJtzBaGEumZ+8Zdeb5QCAJOTOeDJAYfvbmdUCsT4gEiq5fBPrzPCzCVg0iVIJlCLLPEs7vfYBMHUJxndJmn8p/b394zwIVgBrzJSJ1SUMmQfSLi1ChFd888Hhnpy40Mo2MMpTe4Ebya/YkUmO0KmytJ6enwPgf+bEeZYjX/mTU0CFj8zUiQUbfUro+8MA8VyeLcRbsENRvIvp/ggzUuZJ7g5F/gU6DmFxRC5IULMVswDBv6ukqGpFdJPDMwT8z2z5WsoPD1lRmgiWO2mJk7BgxrTZg0Atgb5mr6teRdwAJjU15uqU9pO88gcvc5I+2vpNgSguWRU8s21GNA27zzGCHMXf4znPa6NNWPe610AeCejLw0NzuFxto8lEwX/AMTz+NXLV+bLTO25aMUr7cuQwfBRI1baLR8tHmlUcMs296hbfF4iHICXZwxYpe/T8akvfE5/eNg4/4HRKWi8sfkiusqTU0qz5rgEQDnhVDoGHPkQlJnSgPglPFN5eVqJt9Z+YhJ+DKtyjak2i7EHFh4LthzRJUVKaRuXYrGvdETnR9pEC/58+G51lk0JgN5+HwNXDQBlFo1pUY0V7UeA5CU7MVzM9NHCnZrerBlZJKtX5wY99NZR7XZFg2Y3rcL7rqv8P3pZAqA3IdhTSoGR+oBjeznSflP7PALD3smz7B+h6fcC/abSYZX3BECvAXPaAvl5i9GNRl2d05uYJPG3SYZNCYDSXPQnBy6vIcfKYFZ+2qik/VTX3qYKDMUEPRczVP7J3gtZTmZT+SE5WmYogR0i4acUiPFBpVIso2qhLQHQ27spAKX9iG6xh8ND9Iy9If9gVvJ5KctBK0upT9TNxN7K6D5G9zcipSkNHMh/LYXjBfxzMZEEwF7Mau+iMHcEY1jskIlXZOatSnB3KoTyeIQROwLaglMENFUUYOWhRkT42b57tpEyrpSCx/bOhdMbNcKuCWXMXkiVCQH4EHI9vkyt9uPdWqDqygHMG4XNU2b4lSpnOvv5y2WN1M4vwsV+0NrHp4atqctrnF4AyHj8NRF/sr9/ZNRzQEbamUXiTUaYx8gb8ozMpYs6MsRzlREJ4DzywX0iAp+jyG4KRNYAEB9Jgi5wpbmGYyAFVDRYs6FOHXNC5zpwmTONve9aMn1hNqtaqVwOGD7IkPu+5rOc1VDzXq6MB+DqCN3STpz4ez431UfsPW+nFvVULrHEAOz7jLiX5s9GN/m7NQD0QqZV1Ws/VjKlPsF7lZH2qxWEOQD0Gi4XnfX9KZmWYuySqCFmxy1XCtx/7f1We2J+4amZ25VdP7vXBT4lDXgt7QE4tVgr8MTCyvlKXKRRBVQzSelm/A+tQo6i+Ez73Ib9HO3HTM4BoLSWB7ykoRQ4SqVmDQB7k0AfFKuZ2976f2R/vEzlAObzVqf2qaV4GEsOyBfGWDNJflKJrPGAcp8868PNKkPks1b7zQWgHHtWmZQZfjGoybAXAHuIGq4VQM1DKfC0tp1zex+LCsVCLGFse2HqZIjnh0/hLMrfXABSIaFkOulPQqcb9pSp9f3UeXxNnPOagIRMzNRn9Novpx1zgrMmCNObIGrxmeNL9zaGI/rjTfdc8Mrvd9e4KvIDSwHA4kY8zPBCjcbhSY+O+AGoTE2mvGe2zqWl2xjphIztxSjaRMCIlSynHacAuGT/sbcoaNNN4iOQcFCbAoxOz6cHtr1ZWbLqvClbDLDVaMBUq+TUak2ZEm9lVpaCJn6Anhm8z4P2BUy1WgDTgnzQEvBz/e8tCipBqlmlS/Ox9vtbWAXPXVtJ4/dJ8p9a0HPmpzdPxwCqOjE/sfyQKW1HTAK2BoBe44wdnEy15JLDlQJWaf9KQPXM4F0uCgJ82pOUEPId9v3YNRRzg0V+AnuLgmo7qEmWxkrh/1iHAORaxW8k4/Kym1MuPpFAAAVoRL/T6DWu0fVGX3dy6OeCti4kjNQAkP5qYqecypoypTnFD/TR1Vx5aR3Z1zpcixa7l5EyFVh5OLv1cqMxv/Ko/NMSH5Z+31sWTIskhaW8yL1HNkv6+Gyv1GfD9HylkTK9ABPXW3zSKL1jhgCYvyFA1ohAq6yaCznHtQAE2Zh1U5qNyV97tUDN8SAJGSbmN41eYHRpWFl8MAhmPd/I5x+mzNd+5hLzs6VgtKjLWyolP6VFe+dUh8DCmHQLwMsGmX+v/UtEnwcFxEKPlmNDncUei4v34b/fuPf7gZikpGJeJou1AETrsHpM3QEJSDlKtPSeSAaoEO7URqffZAeEfiVK92Au2fdTp7Zh3B6Hf/cQVm9K1c7rHv06hTbgHVds+IuH0YbcDs71E3Jr+BeLikwWH3jE70vBle5HZxVB7UShpv9iNHWdH2VaZHSglXL3VPqJZHB/NcrlE8ovK12XAAOvMTqXUxCa8NgDXA55eMjz60TWx3y+kqzx3h2nlEAtAJcPadmbcmZzd/4vq/HiW4pWzd0qadH2VnUIgLnoL4LCz3b5JGiCU5wm6fbKvq0Y1VO9vQJQod+trg4E4Nj22Obn8sj/9YEy+EgC8HVG6b4o5je+75KI9bnw7PBx9ApAGIOWwpFtfYkudba87PfwSRw6IACmKXWK8nnNqC2f2r3SXsZ4dv3oGYAwWwcZW92QhuD9x6jkH/Y+0ZftJ1mHtT2TJjIoMOU355XX2MOGfe+83rR/vQNw08GfaOUCT7onO3a3jcLhSnDw+16lpIcTZdHpdDsAeDpzpZ6miQh8jpZLT6iovHw97XXq/dpc2dPj0An1OAB4QpM1dFXXJShApb+vte/ToJL2VZWNgZ/Ivha/BcFnLx3+7vl6iNOboRk9DgDOYFZHRfV7G3SJn1sj9zAHIvl/ZHdQjguGCECR2cFnTzZqHeTqiE39dyUA2P8cremhv0RW5zP9Z8rqWNPGVu8SaOIXdLlJjMXF51CyAL3KiH+7/N2/WqYEAGs5dbrlcllDfJZme/Q0QvzZPw7g02+T+CASv/SrtLHiodeeBpb2JQDY8+xE39LrHbhZjKNlaO7fGJFkwF5mixTIQ7gdADyE7dHoDA7opAJA+5cR2m+rFMUZ3WpTNADYho9Ry3Yc8Bd+nV3mTgBwO8GJmttwwB8xW3J5cptebFRLAHAjxka1zTjg/cBzuDryAmMCgM3kJCraiAO5O1k2amr/agOA+/M8WqzngLJ8CMSwHVG85q++6j5KBgD7mIfoxeUcwPTkipNLRnc20r0suu+GkzJ8frJbEAw5ABii3yMHdMkylx+x5eD9QF3zdxbnOgOAPYrfzbNP8vVImePh1gKffqb9QO68UZman73rmpsBwK6n52bVOZ1zZNC5G6r9lZNNfp+9B+4GAHuYheiDOECO6tTvJXKYmOek/T4/3QHAEP7gwIEcCAAeyPxoOjgQAAwZCA4cyIEA4IHMj6aDAwHAkIHgwIEcCAAeyPxoOjgQAAwZCA4cyIEA4IHMj6aDA/8H9VMBdR60QfQAAAAASUVORK5CYII=\" alt=\"W(u) = integral(exp(-x)/x,{x,0,infinity})\" style=\"width: 112px; height: 35.5px;\" width=\"112\" height=\"35.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-8px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"u_w = S_y r_w^2/4Tt\" style=\"width: 89.5px; height: 22px;\" width=\"89.5\" height=\"22\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 42.15px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 21.075px; text-align: left; transform-origin: 363px 21.075px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 356.708px 8px; transform-origin: 356.708px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eCompute the number of wells by dividing the demand by the initial estimate of the pumping rate and rounding up to the nearest integer: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAALEAAAAoCAYAAACxQBtOAAAIdklEQVR4Xu2cx+slRRDHd/8AsycRD64eBEUxrGA46MEI4kUM60FQjOBBUNd0MoDhJBhREFFU1MuCawIFExhRRBQMeBBPZv8ArQ90QdFMh5rpNzvD9oPit/umZ6a66luxu9/WLf3TJbByCWxdOf+d/S6BLR3EHQSrl0AHcVqFL8ql/ROXb5Xvv1qZ9g8Sfh8RumSBfJ8nPN2Y4Osd+f6BHM8dxGnp/CeXHhP6eWDI0/Ld7wsEQ46la+TizUJHLJDvY4Wnswf4OjN8d04H8TitAeLThT4Yd/vi7vpUOHpK6InFcZZm6Ba5BJAng/i0gXd8m/FEWNU+A/f8K9+tKQS3BjHh/CgjlzmN43B5749C24R+GglinnFIuHcuXTYF8SnC/P1m8tdmLBoQo6xHhQ4I99wnfwnBYwU4Uu6TbmsBYhR/pdDFAUBvBo62h7/k3Q/NIJd75R1nCZ3klAgO7DIhzaM/CfcT+jGKJ4NeN5VaNQOxzhtQXhf+gzJ2COWYR3C3CwHgO5zCW8LwKSDG694U5s9cLhUCsPrh+t1Bnn/K3wuENumZf5Dn3xnxkJMxxoe+FaxXRPxx/SWhE4U+E8JIN+GgNgpiBHCcUC49QGlMrjRuCYAd4mEsiIlE5J4oGG91slDK2AEXIZ5xeMpNAAF+vhQ6OMOHnT+dgueEiKIANdXN0BSFexl3Q+XzPfpuDmIVuDJB5X59hqM/5Bq0xGq4RpBjQAxg3g0A4B2lHJSOweOBmU1FLDzqgRkwWlkA2BfCFxhWSXfqqLjlfKHdNYJ1jGkKYrU6+qOaGxMGj0xYn1r/phTjkMPooV4QkyJ8HIDLS+MUYogRcs73DWhyXnvsRHAkl1cATHWm7ykZIOMAmeIh57XH8t4UxOoxSA34t+bGqQJPx6+5ReUF8RsiF+11kifWFFEWxCi6deqlqQGeOPfBAL8XsoV4TR1jQVxKncYAuSmICRvkbAjDCh5l0cOLcz4NM2teTPGA2IZhDxhjELcOyegBT5xL++DXFu1E2BLoFZAWxHxXm3fXAropiBEEAlFh0DincOEz5G0Z/5bQ2CXOVK+5dvI6bkrF7wGxrRdqvTA82pw4JUvvnHU83vU3oZJ3twUa93pSQJsTc29rp9UMxOotbI5nPU+cC+n4XC+5pBgbmktjU9c9HmXoGbUgjr2wZ97WA7b2ZPB1j1CpOIt5KIHeymqs8dbqtBmINWTEiT7elhwKsOCVtT2UGl/LOOPoMZ/guWFg7F/y3dhIwONqQRx7I09ItSCo6QZ4REK0fFkou3lGrqseebYnisQevNSt8vCuY5uBGGEA1tiidTGDF1rvgxdlbMkDjJnUnPfUgtiCgEWg7Dq/mcAmQVC7zBx3JOg+lUCvU4gjUE03xqu/JiDWvGrIyqwStMCDSfIwT17lndhc42tAPCWfjIsiTxgvyYBnXyRU6pDEQPQUljblm5q6pebTBMS0aF4TSk3OhlIKvH3D+DW31lSgNSCe0l2wqURtj5VCcD8hdnbl9tnWLjOP7S7ExuupA0oGaK83AbEm/ak8zyoRRZDX0UOubdGkJrSW7sRYEMcesGZhQWWlHjDlKDzLzGNBbIvB1rl8cxBj0YSKXFiKK1QmNaWgYhJr6k7gsfVTE47jlT2vF6NGAfSp1VLPMnNshDVFaZxHt0yDYqc22RPXLh3Hvc4WCf6auhO2Z14DSOvFvBW91ii5AlIXN+yuuVTE0+fp9RIgGY+D0TWCFrrOpReTQazgLHmXeMnSExo9+dHcY2tyYnjK9cxjnu1YL4Dtu1LGUrvMbPmyRlUyQjt20wCGx8kg1jShJsTo5Dx9xrlB6X1fLYjj9CeVq9r8sxYAREOcCTWGHlplf0bKY6IHPqVlZisL64RSW0L1kClba0kvzxAa2obLODYxcehzu5BNedSL6zYFjfQ5w5kEYpsi1HQa7C632j6jF1Rzj/eAOFbybcLsN4HhQ+UvK2dEKM++W/XaqmTNX1MbbTQ1qNFXLMt4DzQg/CcM4lTPTiHWCmr7yEN7yfU7jdQKarYnpDYbjQZxnI+y8vWFUAmceAEOIa7pHF3OMDwg1ucAtHOFONGCx+I4D/Jjj/HbQrWb3hXAtt+uIE6lIbXLzLk584wLhfC4GAvRGJ7fC/zXHkPSqKMRx260t+kmWHtdKLXHZTSIc5Pcm66NATHe5XkhQr6nNkDJ3wXAENVIyyjQ7P5ijY6pVISC63Ohmi2UKT3ikV8NFz0rroAfI1WQ2yjyinwPb3hcjFsjRc3vYHQQT7Q4L4jtuTPSBrwvKcWvAZwpduJDnLqcH4du7YIM1Si1y8w5keDpdwmRNhABvhb6RSh3sl2NNjYeGzXI558VOlqIDfRqhACb3D0XnTqIZwSxDZdDr8WzAkLCMsDgQ655tRBph92TrQW1Ld4UFKnCGS99lVBpmTklknjRIx5HSw+gfihErszK7KlCLGzZLbp6n83f4RnPbHdDHiP/x0hKbcAO4hlArKeaUSa56t/hnSwLD/2iTczSUKGkiye2ANJjT3hIckhAtNs8DOA/KOT9YRQ91UzqwPF7PixrcwBCe8EpMWJ8qWNPdlla52GBTWpR00HpIJ4BxLyCPDLXbiKMHi+kLTL1aABzqFDS1UoAi+d7WEiPxfMd4f4uc69nmTkWCUYIaIdCOkCkTXaYkP6cFPezZ+MjodKBA4zRGqkC29Oh6SBuAOLSCtbEVwzejrIBK6AH8PzoDL+oRD75jJAtoHgAisZIpi71t54L3SprbDyf9MFztF9/aDC7vbX1cZLWgtiTz0t52D3J09C7AT1pTG37a2n8l/hhftnWZAdxSYT9+uIl0EG8eBV1BksS6CAuSahfX7wEOogXr6LOYEkCHcQlCfXri5dAB/HiVdQZLEngfx8SS0f3YPfQAAAAAElFTkSuQmCC\" alt=\"ceil(Q_d/Q_w)\" style=\"width: 88.5px; height: 20px;\" width=\"88.5\" height=\"20\"\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 21.7167px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.8583px; text-align: left; transform-origin: 363px 10.8583px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 74.675px 8px; transform-origin: 74.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSet the pumping rate to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"Q_0 = Q_d/N\" style=\"width: 73.5px; height: 20px;\" width=\"73.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 206.542px 8px; transform-origin: 206.542px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eArrange the wells so that they are equidistant from the central well.\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 43.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 21.7167px; text-align: left; transform-origin: 363px 21.7167px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 73.9083px 8px; transform-origin: 73.9083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eDetermine the distance \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eR\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 261px 8px; transform-origin: 261px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e between the central well and others so that the total drawdown at the central well is \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"s_max\" style=\"width: 25.5px; height: 20px;\" width=\"25.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. In other words, add the drawdown from the central well to the drawdown from the other wells. If \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg 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alt=\"u_R = S_y R^2/4Tt\" style=\"width: 91px; height: 21px;\" width=\"91\" height=\"21\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 19.4417px 8px; transform-origin: 19.4417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, then \u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e\u003cdiv style=\"block-size: 37.9px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.95px; text-align: left; transform-origin: 384px 18.95px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 23.3px 8px; transform-origin: 23.3px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg 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style=\"width: 227px; height: 38px;\" width=\"227\" height=\"38\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 174.133px 8px; transform-origin: 174.133px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e Write a function to design a well field using this 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\" data-image-state=\"image-loaded\" width=\"389\" height=\"374\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 118.5px 8px; transform-origin: 118.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e*http://www.aqtesolv.com/neuman.htm\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [N,Q0,R] = wellfield1(Qd,smax,rw,K,Sy,b)\r\n % Q0 = pumping rate (m3/d)\r\n % N = number of wells\r\n % R = distance between central well and other wells (m)\r\n N = randi(10);\r\n Q0 = round(Qd/N,'up');\r\n R = b;\r\nend","test_suite":"%%\r\nQd = 7000; % Demand (m3/d)\r\nsmax = 3; % Maximum drawdown (m)\r\nK = 50; % Hydraulic conductivity (m/d)\r\nSy = 0.2; % Specific yield \r\nb = 30; % Initial saturated thickness (m)\r\nrw = 0.3; % Radius of the well (m)\r\n[N,Q0,R] = wellfield1(Qd,smax,rw,K,Sy,b);\r\nQ0_correct = 1400;\r\nN_correct = 5;\r\nR_correct = 189.4;\r\nassert(abs(Q0-Q0_correct)\u003c1e-1)\r\nassert(isequal(N,N_correct))\r\nassert(abs(R-R_correct)\u003c1e-1)\r\n\r\n%%\r\nQd = 6000; % Demand (m3/d)\r\nsmax = 2.5; % Maximum drawdown (m)\r\nK = 60; % Hydraulic conductivity (m/d)\r\nSy = 0.22; % Specific yield \r\nb = 20; % Initial saturated thickness (m)\r\nrw = 0.25; % Radius of the well (m)\r\n[N,Q0,R] = wellfield1(Qd,smax,rw,K,Sy,b);\r\nQ0_correct = 857.1;\r\nN_correct = 7;\r\nR_correct = 297.6;\r\nassert(abs(Q0-Q0_correct)\u003c1e-1)\r\nassert(isequal(N,N_correct))\r\nassert(abs(R-R_correct)\u003c1e-1)\r\n\r\n%%\r\nQd = 10000; % Demand (m3/d)\r\nsmax = 4; % Maximum drawdown (m)\r\nK = 80; % Hydraulic conductivity (m/d)\r\nSy = 0.18; % Specific yield \r\nb = 25; % Initial saturated thickness (m)\r\nrw = 0.4; % Radius of the well (m)\r\n[N,Q0,R] = wellfield1(Qd,smax,rw,K,Sy,b);\r\nQ0_correct = 2500;\r\nN_correct = 4;\r\nR_correct = 117.6;\r\nassert(abs(Q0-Q0_correct)\u003c1e-1)\r\nassert(isequal(N,N_correct))\r\nassert(abs(R-R_correct)\u003c1e-1)\r\n\r\n%%\r\nQd = 12000; % Demand (m3/d)\r\nsmax = 3; % Maximum drawdown (m)\r\nK = 140; % Hydraulic conductivity (m/d)\r\nSy = 0.2; % Specific yield \r\nb = 25; % Initial saturated thickness (m)\r\nrw = 0.4; % Radius of the well (m)\r\n[N,Q0,R] = wellfield1(Qd,smax,rw,K,Sy,b);\r\nQ0_correct = 3000;\r\nN_correct = 4;\r\nR_correct = 78.2;\r\nassert(abs(Q0-Q0_correct)\u003c1e-1)","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2022-12-23T22:33:37.000Z","deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2022-12-23T00:54:17.000Z","updated_at":"2026-02-12T15:43:55.000Z","published_at":"2022-12-23T00:58:29.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA well field provides water for a community. The design of a well field involves a goal to meet a specified service demand \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Q_d\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eQ_d\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e (i.e., volume of water per time) with the constraint of lowering the water table by no more than \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"s_max\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es_{max}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, the maximum drawdown. Inputs to the design are properties of the aquifer (the hydraulic conductivity \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"K\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eK\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, the specific yield \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"S_y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eS_y\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and the initial saturated thickness \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"b\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eb\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e) and the radius \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"r_w\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003er_w\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e of the well. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe Gupta/Chin method for designing a well field has the following steps:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCompute \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Q_w\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eQ_w\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, an initial estimate of the pumping rate, such that the drawdown at one well (i.e., at a distance \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"r = r_w\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003er = r_w\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e) is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"s_max/2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es_{max}/2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Compute the transmissivity to be \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"T = K(b-s_{max}/2)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eT = K(b-s_{max}/2)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Evaluate the drawdown at a time \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"t = \\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003et =\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e 1 year. Realize that for small values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"u = S_y r^2/4Tt\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu = S_y r^2/4Tt\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e the unconfined well function* can be approximated and compute the pumping rate from \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"s_max/2 = (Q_w/4 pi T) W(u_w)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\frac{s_{max}}{2} = \\\\frac{Q_w}{4\\\\pi T} W(u_w)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"W(u) = integral(exp(-x)/x,{x,0,infinity})\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eW(u) = \\\\int_u^\\\\infty \\\\frac{e^{-x}}{x} dx\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"u_w = S_y r_w^2/4Tt\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu_w = S_y r_w^2/4 T t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCompute the number of wells by dividing the demand by the initial estimate of the pumping rate and rounding up to the nearest integer: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"ceil(Q_d/Q_w)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eN = \\\\lceil Q_d/Q_w\\\\rceil\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSet the pumping rate to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Q_0 = Q_d/N\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eQ_0 = Q_d/N\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eArrange the wells so that they are equidistant from the central well.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eDetermine the distance \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"R\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eR\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e between the central well and others so that the total drawdown at the central well is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"s_max\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es_{max}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. In other words, add the drawdown from the central well to the drawdown from the other wells. If \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"u_R = S_y R^2/4Tt\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu_R = S_y R^2/4Tt\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, then \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es_{max} = \\\\frac{Q_0}{4\\\\pi T}\\\\left[W(u_w) + (N-1) W(u_R)\\\\right]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e Write a function to design a well field using this method.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"374\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"389\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc 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an ODE: equation C","description":"Write a function to solve the following ordinary differential equation: \r\n\r\nwith and . The parameter is a constant. The function should return the values of at the specified values of .","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 118.583px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 59.2917px; transform-origin: 407px 59.2917px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 209.417px 7.91667px; transform-origin: 209.417px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve the following ordinary differential equation: \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 37.3333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.6667px; text-align: left; transform-origin: 384px 18.6667px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"(1-x^2) y\u0026quot; - x y' + a^2 y = 0\" style=\"width: 179.5px; height: 37.5px;\" width=\"179.5\" height=\"37.5\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.25px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.125px; text-align: left; transform-origin: 384px 21.125px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14.3917px 7.91667px; transform-origin: 14.3917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewith \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(x0) = y0\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(x1) = y1\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 51.725px 7.91667px; transform-origin: 51.725px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The parameter \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 169.967px 7.91667px; transform-origin: 169.967px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a constant. The function should return the values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ey\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 50.95px 7.91667px; transform-origin: 50.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at the specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = solveODEC(x,a,Xbc,bc)\r\n% y = values of the function\r\n% x = independent variable\r\n% a = parameter in the equation\r\n% Xbc = [x0 x1], values of x where the boundary conditions are specified\r\n% bc = [y0 y1], values of the function at x0 and x1, respectively\r\n\r\n y = f(x,a,Xbc,bc);\r\nend","test_suite":"%% \r\nx = [-1/3 -1/4 0 1/4 1/3];\r\na = 1;\r\nXbc = [-1/2 1/2];\r\nbc = [2 4];\r\ny = solveODEC(x,a,Xbc,bc);\r\ny_correct = [2.599319657044238 2.854101966249684 3.464101615137754 3.854101966249684 3.932652990377571];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%% \r\nx = [-0.6 -0.2 -0.05 0.12 0.2];\r\na = sqrt(3);\r\nXbc = [-0.7 0.3];\r\nbc = [0 1];\r\ny = solveODEC(x,a,Xbc,bc);\r\ny_correct = [0.237055225759061 0.877546669526703 0.995431932094838 1.046546024897035 1.039090864471318];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%% \r\nx = [-0.8 -0.6 -0.3 -0.15 0.1 0.25 0.375];\r\na = 2;\r\nXbc = [-0.8 0.4];\r\nbc = [-1 3];\r\ny = solveODEC(x,a,Xbc,bc);\r\ny_correct = [-1 1.68647951290722 4.138417234449975 4.687455882945745 4.630189304757032 4.024530246664777 3.199461161409147];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%% \r\nx = [-0.7 -0.45 -0.2 0.05 0.3 0.55 0.8];\r\na = pi;\r\nXbc = [-0.9 0.9];\r\nbc = [-1 1];\r\ny = solveODEC(x,a,Xbc,bc);\r\ny_correct = [1.764854605944604 2.706629373469937 1.6090059417224112 -0.4259046519843118 -2.225041323571773 -2.630850806938173 -0.6162122684969365];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%%\r\na = 5;\r\nXbc = [-1/2 1/2]; \r\nbc = [-1 1];\r\ny = 1;\r\nx = fzero(@(z) solveODEC(z,a,Xbc,bc)-y,0);\r\nx_correct = 0.104528463267653;\r\nassert(abs(x-x_correct)\u003c1e-13)\r\n\r\n%%\r\na = 5;\r\nXbc = [-1/2 1/2]; \r\nbc = [-1 1];\r\ny = 1;\r\nx = fzero(@(z) solveODEC(z,a,Xbc,bc)-y,0);\r\nx_correct = 0.104528463267653;\r\nassert(abs(x-x_correct)\u003c1e-13)\r\n\r\n%%\r\na = 2*randi(4)+1;\r\nXbc = [-3/4 3/4]; \r\nbc = [-2 2];\r\nx = rand-3/4;\r\nym = solveODEC(-x,a,Xbc,bc);\r\nyp = solveODEC(x,a,Xbc,bc);\r\nassert(abs(ym+yp)\u003c1e-13)","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-05-25T04:18:45.000Z","updated_at":"2021-05-25T04:23:56.000Z","published_at":"2021-05-25T04:23:56.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve the following ordinary differential equation: \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"(1-x^2) y\u0026quot; - x y' + a^2 y = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e(1-x^2)\\\\frac{d^2y}{dx^2} -x \\\\frac{dy}{dx}+a^2 y = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewith \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(x0) = y0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(x_0) = y_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(x1) = y1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(x_1) = y_1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The parameter \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a constant. The function should return the values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at the specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60748,"title":"Extend the digamma function to negative arguments","description":"While solving one of Ramon Villamangca’s problems, I needed the value of , where is the digamma function. However, MATLAB’s function psi does not work for negative arguments. \r\nWrite a function that extends the digamma function to negative arguments. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 72px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 36px; transform-origin: 407px 36px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWhile solving one of Ramon Villamangca’s problems, I needed the value of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"57.5\" height=\"18\" style=\"width: 57.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"32\" height=\"18\" style=\"width: 32px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the digamma function. However, MATLAB’s function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003epsi\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e does not work for negative arguments. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that extends the digamma function to negative arguments. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = newpsi(x)\r\n y = -psi(abs(x));\r\nend","test_suite":"%%\r\nx = -1/2;\r\ny = newpsi(x);\r\ny_correct = 0.0364899739785765;\r\nassert(abs(y-y_correct)\u003c1e-13)\r\n\r\n%%\r\nx = -1/3;\r\ny = newpsi(x);\r\ny_correct = 1.68176558421341;\r\nassert(abs(y-y_correct)\u003c1e-13)\r\n\r\n%%\r\nx = -1/4;\r\ny = newpsi(x);\r\ny_correct = 2.91413912021353;\r\nassert(abs(y-y_correct)\u003c1e-11)\r\n\r\n%%\r\nx = -1/5;\r\ny = newpsi(x);\r\ny_correct = 4.03499143329386;\r\nassert(abs(y-y_correct)\u003c1e-11)\r\n\r\n%%\r\nx = -2/7;\r\ny = newpsi(x);\r\ny_correct = 2.31981855966050;\r\nassert(abs(y-y_correct)\u003c1e-11)\r\n\r\n%%\r\nx = -3/8;\r\ny = newpsi(x);\r\ny_correct = 1.21395790209010;\r\nassert(abs(y-y_correct)\u003c1e-11)\r\n\r\n%%\r\nx = -4/9;\r\ny = newpsi(x);\r\ny_correct = 0.537183359660484;\r\nassert(abs(y-y_correct)\u003c1e-11)\r\n\r\n%%\r\nx = -5/11;\r\ny = newpsi(x);\r\ny_correct = 0.444812910020638;\r\nassert(abs(y-y_correct)\u003c1e-11)\r\n\r\n%%\r\nx = -17/13;\r\ny = newpsi(x);\r\ny_correct = 2.77268958155960;\r\nassert(abs(y-y_correct)\u003c1e-11)\r\n\r\n%%\r\nx = -37/19;\r\ny = newpsi(x);\r\ny_correct = -17.9247506862983;\r\nassert(abs(y-y_correct)\u003c1e-11)\r\n\r\n%%\r\nx = -exp(1);\r\ny = newpsi(x);\r\ny_correct = -1.39770931071902;\r\nassert(abs(y-y_correct)\u003c1e-11)\r\n\r\n%%\r\nx = -pi;\r\ny = newpsi(x);\r\ny_correct = 7.88595238538549;\r\nassert(abs(y-y_correct)\u003c1e-11)\r\n\r\n%%\r\nx = -sqrt(73);\r\ny = newpsi(x);\r\ny_correct = 1.76552281819428;\r\nassert(abs(y-y_correct)\u003c1e-11)\r\n\r\n%%\r\nx = -sqrt(273);\r\ny = newpsi(x);\r\ny_correct = 2.61015605350263;\r\nassert(abs(y-y_correct)\u003c1e-11)\r\n\r\n%%\r\nx = -sqrt(7073);\r\ny = newpsi(x);\r\ny_correct = 13.9912915572908;\r\nassert(abs(y-y_correct)\u003c1e-11)\r\n\r\n%%\r\nx = 1;\r\ny = newpsi(x);\r\ny_correct = -0.577215664901532;\r\nassert(abs(y-y_correct)\u003c1e-11)\r\n\r\n%%\r\nx = sqrt(7073);\r\ny = newpsi(x);\r\ny_correct = 4.426062993434546;\r\nassert(abs(y-y_correct)\u003c1e-11)\r\n\r\n%%\r\nx = -[293/37 263/47 233/67 223/97 193/107 173/127 163/137];\r\ny = arrayfun(@newpsi,x);\r\ns = sum(y);\r\np = prod(y);\r\ns_correct = -0.500409630494963;\r\np_correct = 1714.480727245714;\r\nassert(abs(s-s_correct)\u003c1e-11)\r\nassert(abs(p-p_correct)\u003c1e-8)","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2024-10-13T04:37:44.000Z","deleted_by":null,"deleted_at":null,"solvers_count":6,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2024-10-13T04:28:19.000Z","updated_at":"2025-10-02T13:47:28.000Z","published_at":"2024-10-13T04:37:44.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWhile solving one of Ramon Villamangca’s problems, I needed the value of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\psi(-1/2)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\psi(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the digamma function. However, MATLAB’s function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003epsi\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e does not work for negative arguments. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that extends the digamma function to negative arguments. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":46117,"title":"Test approximations of the prime counting function","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 162.817px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 81.4083px; transform-origin: 407px 81.4083px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 90px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 45px; text-align: left; transform-origin: 384px 45px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/241\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problem 241\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 61.4583px 7.79167px; transform-origin: 61.4583px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, which is based on \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"http://projecteuler.net/problem=7\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eProject Euler Problem 7\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 73.1083px 7.79167px; transform-origin: 73.1083px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, asks us to identify the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eN\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 91px 7.79167px; transform-origin: 91px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eth prime number. That is, the problem seeks the inverse of the prime counting function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAD4AAAAlCAYAAAAa0FocAAACX0lEQVRoge2YXZGDMBSFPw84wAAGUFAFdYADHNRCNSChHmoBDVjoPoSzvaVJubCkywNnhtmZpHv/cnJuEjhw4MCBPKiAYiNb5Ya2suIMXDe0VwG38e9usXXSQgn07DT5EyG4XLSsMttfhRIYgDqznwvQZfaxCFfCPsyNglDg0xd8zaIEHuRfbeFKoPy/49uBVHy30EkM5FHyrD5LQgtqHV+sldSE6p8X+KzH31t7JdB88DPFjZUsKwgVeyz4yoidFh/trsA9Ya+djA8JXzG/i1pbMQbREyonh7fxU4C9GUvRSsXzBFBMfOn/b4REL2Z+TrW9BX/BjUAreAqFbUXNONY6bT2cfqX+D560tj1ZyQzMF/KEr0C/KHjdQ1qxxoz145h3r3kTP/NM/Mz7KUy2POcBaYtncd5gqac9JQZ4hWNJ4iryMNqf0nTgfRFS+FPiorStsPbZxWljSeJKLBZwbeY8TFvTTX4hStsgFJxXNDp8iYtJKTa1H+ZiWCVuEK+wBMMjLtMA5lbJtqzYKqmTeA8lsjfX9t4git7NWBcZu8wY96qrEhsic1ZrZGcuoS5hK4mSZ4LT6ov6olvDaxFiUNCfRMYmFltRqb2YVjn89iy4nlp6T1eWyZw+zx7qZgIVK1L2LNP02PBp60gv3MJm++gQMW6Z4G0r1m6KnraNxbTDHp09jxktC2kOofpNIsiC54lqqWj0pFugLkGpFSrG+QafqPas7N85UOO7XPwVDTt5hLDI/R6md71dvrR2+LVhKe6sPKl9A7rnbx1gDptZ0LLd+/eJHbyvHThw4MC/4wdzuO6gMG8oWgAAAABJRU5ErkJggg==\" alt=\"pi(n)\" style=\"width: 31px; height: 18.5px;\" width=\"31\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 185.15px 7.79167px; transform-origin: 185.15px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, which provides the number of primes less than or equal to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.8833px 7.79167px; transform-origin: 17.8833px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://mathworld.wolfram.com/PrimeNumberTheorem.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime Number Theorem\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 87.9px 7.79167px; transform-origin: 87.9px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e gives approximate forms of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"pi(n)\" style=\"width: 31px; height: 18.5px;\" width=\"31\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 29.5583px 7.79167px; transform-origin: 29.5583px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e for large \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 96.8583px 7.79167px; transform-origin: 96.8583px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Two such approximations are \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"n/ln(n)\" style=\"width: 52px; height: 18.5px;\" width=\"52\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 27.225px 7.79167px; transform-origin: 27.225px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and the \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Logarithmic_integral_function#Offset_logarithmic_integral\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"perspective-origin: 77.675px 7.79167px; transform-origin: 77.675px 7.79167px; \"\u003eoffset logarithmic integral\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.79167px; transform-origin: 1.94167px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"Li(n) = li(n) - li(2)\" style=\"width: 126px; height: 18.5px;\" width=\"126\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24.8917px 7.79167px; transform-origin: 24.8917px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-8px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"li(x) = integral of 1/ln(t) from t = 0 to t = infinity\" style=\"width: 122px; height: 27px;\" width=\"122\" height=\"27\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 18.6667px 7.79167px; transform-origin: 18.6667px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e (See \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46066-evaluate-the-logarithmic-integral\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problem 46066\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4.275px 7.79167px; transform-origin: 4.275px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e).\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63.8167px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.9083px; text-align: left; transform-origin: 384px 31.9083px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 160.642px 7.79167px; transform-origin: 160.642px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eTest these approximations by computing two ratios: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"r1 = [n/ln(n)]/pi(n)\" style=\"width: 128px; height: 20px;\" width=\"128\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.79167px; transform-origin: 15.5583px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"r2 = Li(n)/pi(n)\" style=\"width: 100.5px; height: 20px;\" width=\"100.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 57.95px 7.79167px; transform-origin: 57.95px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Do not round the approximations to integers. For \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"n = 100\" style=\"width: 51.5px; height: 18px;\" width=\"51.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 257.5px 7.79167px; transform-origin: 257.5px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, you will find that the first approximation is about 13% low and the second is about 16% high. However, for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"n = 10^8\" style=\"width: 49px; height: 19px;\" width=\"49\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 216.658px 7.79167px; transform-origin: 216.658px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, the first approximation is 6% low and the second is only 0.01% high. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [r1,r2] = primeCount(n)\r\n r1 = (n/ln(n))/primepi(n);\r\n r2 = Li(n)/primepi(n);\r\nend","test_suite":"%%\r\nn = 1e2;\r\nr1_correct = 0.86859;\r\nr2_correct = 1.16324;\r\n[r1,r2] = primeCount(n);\r\nassert(isequal(round(r1,5),r1_correct) \u0026\u0026 isequal(round(r2,5),r2_correct))\r\n\r\n%%\r\nn = 1e4;\r\nr1_correct = 0.88343;\r\nr2_correct = 1.01309;\r\n[r1,r2] = primeCount(n);\r\nassert(isequal(round(r1,5),r1_correct) \u0026\u0026 isequal(round(r2,5),r2_correct))\r\n\r\n%%\r\nn = 1e6;\r\nr1_correct = 0.92209;\r\nr2_correct = 1.00164;\r\n[r1,r2] = primeCount(n);\r\nassert(isequal(round(r1,5),r1_correct) \u0026\u0026 isequal(round(r2,5),r2_correct))\r\n\r\n%%\r\nn = 1e8;\r\nr1_correct = 0.94224;\r\nr2_correct = 1.00013;\r\n[r1,r2] = primeCount(n);\r\nassert(isequal(round(r1,5),r1_correct) \u0026\u0026 isequal(round(r2,5),r2_correct))\r\n\r\n%%\r\nn = 1e5;\r\n[r1,r2] = primeCount(n);\r\ns1 = floor(1e5*round(r1,5));\r\ns2 = floor(1e5*round(r2,5));\r\nbxo_correct = 59814;\r\nassert(isequal(bitxor(s1,s2),bxo_correct))","published":true,"deleted":false,"likes_count":1,"comments_count":3,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":16,"test_suite_updated_at":"2021-01-03T15:27:20.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2020-08-07T15:57:52.000Z","updated_at":"2025-08-18T01:32:16.000Z","published_at":"2020-08-07T16:33:15.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/241\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem 241\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, which is based on \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://projecteuler.net/problem=7\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eProject Euler Problem 7\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, asks us to identify the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"N\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eN\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eth prime number. That is, the problem seeks the inverse of the prime counting function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"pi(n)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi(n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, which provides the number of primes less than or equal to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://mathworld.wolfram.com/PrimeNumberTheorem.html\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime Number Theorem\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e gives approximate forms of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"pi(n)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi(n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for large \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Two such approximations are \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n/ln(n)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en/\\\\ln(n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Logarithmic_integral_function#Offset_logarithmic_integral\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eoffset logarithmic integral\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Li(n) = li(n) - li(2)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e{\\\\rm Li(n)} ={\\\\rm li(n)} - {\\\\rm li(2)}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"li(x) = integral of 1/ln(t) from t = 0 to t = infinity\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e{\\\\rm li}(x) = \\\\int_0^\\\\infty dt/\\\\ln(t)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e (See \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46066-evaluate-the-logarithmic-integral\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem 46066\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTest these approximations by computing two ratios: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"r1 = [n/ln(n)]/pi(n)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003er_1 =[n/\\\\ln(n)]/ \\\\pi(n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"r2 = Li(n)/pi(n)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003er_2 = {\\\\rm Li}(n)/\\\\pi(n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Do not round the approximations to integers. For \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n = 100\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en = 100\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, you will find that the first approximation is about 13% low and the second is about 16% high. However, for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n = 10^8\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en = 10^8\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, the first approximation is 6% low and the second is only 0.01% high. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":59571,"title":"Compute a sum involving the zeta function","description":"Write a function to compute the sum\r\n\r\nfor , where is the zeta function, the subject of Cody Problems 45939, 45988, and 45997.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 105px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 52.5px; transform-origin: 407px 52.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 111.883px 8px; transform-origin: 111.883px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to compute the sum\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 45px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22.5px; text-align: left; transform-origin: 384px 22.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"137.5\" height=\"45\" alt=\"S(x) = sum(zeta(n+1) x^n) for n = 1 to n = inf\" style=\"width: 137.5px; height: 45px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.1083px 8px; transform-origin: 10.1083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003efor \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"46.5\" height=\"18.5\" alt=\"|x| \u003c 1\" style=\"width: 46.5px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24.8917px 8px; transform-origin: 24.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"34\" height=\"18.5\" alt=\"zeta(m)\" style=\"width: 34px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 157.517px 8px; transform-origin: 157.517px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the zeta function, the subject of Cody Problems \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/45939\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003e45939\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/45988\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003e45988\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.5px 8px; transform-origin: 17.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/45997\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003e45997\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function S = zetasum(x)\r\n n = 1:Inf;\r\n S = sum(zeta(n+1).*x.^n);\r\nend","test_suite":"%%\r\nx = 1/2;\r\nS = zetasum(x);\r\nS_correct = 1.386294361119891;\r\nassert(abs(S-S_correct)/S_correct\u003c1e-12)\r\n\r\n%%\r\nx = 2/3;\r\nS = zetasum(x);\r\nS_correct = 2.554818115119273;\r\nassert(abs(S-S_correct)/S_correct\u003c1e-12)\r\n\r\n%%\r\nx = 3/4;\r\nS = zetasum(x);\r\nS_correct = 3.650237868474732;\r\nassert(abs(S-S_correct)/S_correct\u003c1e-12)\r\n\r\n%%\r\nx = 5/6;\r\nS = zetasum(x);\r\nS_correct = 5.754911840473381;\r\nassert(abs(S-S_correct)/S_correct\u003c1e-12)\r\n\r\n%%\r\nx = 7/8;\r\nS = zetasum(x);\r\nS_correct = 7.811276998394322;\r\nassert(abs(S-S_correct)/S_correct\u003c1e-12)\r\n\r\n%%\r\nx = 8/9;\r\nS = zetasum(x);\r\nS_correct = 8.83072761223029;\r\nassert(abs(S-S_correct)/S_correct\u003c1e-12)\r\n\r\n%%\r\nx = 9/10;\r\nS = zetasum(x);\r\nS_correct = 9.84653927550954;\r\nassert(abs(S-S_correct)/S_correct\u003c1e-12)\r\n\r\n%%\r\nx = 10/11;\r\nS = zetasum(x);\r\nS_correct = 10.85964675709217;\r\nassert(abs(S-S_correct)/S_correct\u003c1e-12)\r\n\r\n%%\r\nx = 11/12;\r\nS = zetasum(x);\r\nS_correct = 11.87068966352595;\r\nassert(abs(S-S_correct)/S_correct\u003c1e-12)\r\n\r\n%% \r\nx = 0.232931374143;\r\nSS = zetasum(zetasum(x));\r\nSS_correct = 1.227707484938568;\r\nassert(abs(SS-SS_correct)/SS_correct\u003c1e-12)\r\n\r\n%%\r\nx = 1./primes(20);\r\ny = sum(arrayfun(@zetasum,x));\r\ny_correct = 3.2640541637441439;\r\nassert(abs(y-y_correct)/y_correct\u003c1e-12)\r\n\r\n%%\r\nfiletext = fileread('zetasum.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert') || contains(filetext,'regexp') || contains(filetext,'switch'); \r\nassert(~illegal)","published":true,"deleted":false,"likes_count":1,"comments_count":2,"created_by":46909,"edited_by":46909,"edited_at":"2024-01-20T17:57:17.000Z","deleted_by":null,"deleted_at":null,"solvers_count":7,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2024-01-20T13:37:10.000Z","updated_at":"2026-03-04T13:56:18.000Z","published_at":"2024-01-20T13:40:38.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to compute the sum\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"S(x) = sum(zeta(n+1) x^n) for n = 1 to n = inf\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eS(x) = \\\\sum_{n=1}^\\\\infty \\\\zeta(n+1) x^n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003efor \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"|x| \u0026lt; 1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e|x| \u0026lt; 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"zeta(m)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\zeta(m)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the zeta function, the subject of Cody Problems \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/45939\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e45939\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/45988\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e45988\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/45997\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e45997\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":59981,"title":"Compute the Ramanujan tau function","description":"The Ramanujan tau function is defined by the relation\r\n\r\nwhere . The first few values of are 1, -24, 252, -1472, and 4830. \r\nWrite a function to compute the Ramanujan tau function.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 125px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 62.5px; transform-origin: 407px 62.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 89.8583px 8px; transform-origin: 89.8583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe Ramanujan tau function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADkAAAAlCAYAAAD4DEFlAAADjElEQVRoQ+2YS6hOURTH750zMWPAgAEDj/IsUcqjZCLlUQYG8hgZUQwNKIYGHhncDIQyIsWAgZTXgFIMUCaMiJjz/9VZtZ32a/m+fQbuPfXvfp279trrv9baa62zJyemwTM5DThOzJD8X6I8RCRPylmXhW9jcNpC6Vgl3PToak0SYx4JlzxGFWTP6f8LhD21OluShOAr4WytMQ45i2QV0VYk8fZGYZ3DcK/oUy2YqsmSFiS3auP7wiLhg9dyh/xqyT4UVpT2aUESD3/ynBkHsb4oacv5zGbMuEkOFUUja/ut0YsXKWeNmyRpOl9YMkJ0vEs/a8HjXOaMk+QcbfRVuCIc9lo6gjwpu12YPUQkKec3hL2Cq1mPQJClDBtnhGTK5iLJdMFivDQrYcg7vV8vMM3QNo4L24QHGcPRCzYLywXS+45woluDnv3CXCHUn1Jp5/KoBKJDR4rkES24WOHhW5Kxhsx5ZMPSEUB+pYBR5jyiQLt5IvzogC6e84EDYiYZyaRczCCMuCpcF253Wk/r79rOsI/BThhmvZDWgUyJpC03eaK1Q7grXBOYkMxwZE9172IE7d1v/Qgd/pds3yCKxxvhgGApR2q9F0qp4yWJYTxEgF73XbCCFWZSzVCBrmdCtF/2SXKIiVRYOPiCOFThUQ/JMFJEAJKcfftSsT1Ljg0jWU0ylhI/9ZKzU/Koh6SRYL9fQn80o/dReGraUbF1lc6PpU3SS4FXiP5uoaSTJW+Fxd3a/pmz48G/S5UamX8qPGE0LTo1HvW0EM44D84L05R35lgiTBqXPraNZLI/57xuacDGoQKm/9icWOxXHbGwqMQqp7UiCh+RLD3m3ORxypG0CSYkCcELEe+bIZxfWk9urDMSqUhZDTAHUAwpTqnPNvQtFealvJEjaR6ytJrSD8r9QSE1tvF+Q25D/c9IxCIVOpbIbBF2CvsyaUv7yA4MOZI2E4YOKp1NS9lUwSATnncKY2NY6Fiit0xgUEhF0VI/W/lLZ/KeNmCK+SIw9dRcSOU+mkMSMcOorNwL0bIgScrmbhfY67WQ/eqpKfepVE+9N0N3SSA3qHv19uVt/CxW4BYkMYY0OibYF8qohPrrqfxcsVQ5shVJjHLfjzo8UX1Th86WJC2im/Sj6n60kqT7wro1Seym4r4USpNLDUfOO6mavLSKKRmCZI3xTWVmSDZ174DKZyI5oLObbvUHImjNJrU0Sj8AAAAASUVORK5CYII=\" width=\"28.5\" height=\"18.5\" alt=\"tau(n)\" style=\"width: 28.5px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 77.8px 8px; transform-origin: 77.8px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is defined by the relation\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 35px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 17.5px; text-align: left; transform-origin: 384px 17.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"172\" height=\"35\" alt=\"Sum[tau(n)q^n,{n,1,inf}] = q Product[(1-q^n)^24,{n,1,inf}]\" style=\"width: 172px; height: 35px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 8px; transform-origin: 21.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"88\" height=\"18.5\" alt=\"q = exp(2 pi i z)\" style=\"width: 88px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 73.8833px 8px; transform-origin: 73.8833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The first few values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"28.5\" height=\"18.5\" alt=\"tau(n)\" style=\"width: 28.5px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 106.183px 8px; transform-origin: 106.183px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are 1, -24, 252, -1472, and 4830. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 98.6583px 8px; transform-origin: 98.6583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to compute the \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://mathworld.wolfram.com/TauFunction.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003eRamanujan tau function\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = Ramanujantau(n)\r\n y = q*prod((1-q^n)^24)/sum(q^n);\r\nend","test_suite":"%%\r\nassert(isequal(Ramanujantau(1),1))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(2),-24))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(7),-16744))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(14),401856))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(22),-12830688))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(51),-1740295368))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(86),411016992))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(147),-427635232164))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(243),13400796651732))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(260),4107578522880))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(300),9458784518400))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(325),14731871253050))\r\n\r\n%%\r\nassert(isequal(Ramanujantau(400),-25171202969600))\r\n\r\n%%\r\nassert(isequal(sum(arrayfun(@Ramanujantau,1:300)),-33462718906943))\r\n\r\n%%\r\nfiletext = fileread('Ramanujantau.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert') || contains(filetext, 'read'); \r\nassert(~illegal)","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2024-04-26T13:15:52.000Z","deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2024-04-25T03:28:46.000Z","updated_at":"2024-04-26T13:15:52.000Z","published_at":"2024-04-25T03:28:58.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe Ramanujan tau function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"tau(n)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\tau(n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is defined by the relation\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Sum[tau(n)q^n,{n,1,inf}] = q Product[(1-q^n)^24,{n,1,inf}]\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\sum_{n\\\\ge1} \\\\tau(n) q^n = q\\\\prod_{n\\\\ge1}(1-q^n)^{24}\\n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"q = exp(2 pi i z)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eq = \\\\exp(2\\\\pi i z)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The first few values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"tau(n)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\tau(n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e are 1, -24, 252, -1472, and 4830. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to compute the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://mathworld.wolfram.com/TauFunction.html\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eRamanujan tau function\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60633,"title":"Sum the reciprocals of polygonal numbers","description":"As explained in Cody Problem 60571, a polygonal number is the number of dots arranged in the shape of a regular polygon. For example, 15 is a triangular number because dots can be arranged in the shape of a triangle with rows of 1, 2, 3, 4, and 5 dots. The number 16 is a square number because dots can be arranged in four rows of four. \r\nWrite a function to sum the reciprocals of polygonal numbers. In particular, compute\r\n\r\nwhere is the th -gonal number (i.e., the th number corresponding to a regular polygon with sides). ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 178px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 89px; transform-origin: 407px 89px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 49.4083px 8px; transform-origin: 49.4083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAs explained in \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/60571\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003eCody Problem 60571\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 9.71667px 8px; transform-origin: 9.71667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, a \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Polygonal_number\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003epolygonal number\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 203.483px 8px; transform-origin: 203.483px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the number of dots arranged in the shape of a regular polygon. For example, 15 is a triangular number because dots can be arranged in the shape of a triangle with rows of 1, 2, 3, 4, and 5 dots. The number 16 is a square number because dots can be arranged in four rows of four. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 258.533px 8px; transform-origin: 258.533px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to sum the reciprocals of polygonal numbers. In particular, compute\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 45px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22.5px; text-align: left; transform-origin: 384px 22.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"74\" height=\"45\" alt=\"y = sum[1/P_n,s,{n,1,inf}]\" style=\"width: 74px; height: 45px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 11px; text-align: left; transform-origin: 384px 11px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 8px; transform-origin: 21.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"23.5\" height=\"20\" alt=\"P_n,s\" style=\"width: 23.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 20.6083px 8px; transform-origin: 20.6083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.775px 8px; transform-origin: 7.775px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eth \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003es\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 74.2917px 8px; transform-origin: 74.2917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e-gonal number (i.e., the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 157.942px 8px; transform-origin: 157.942px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eth number corresponding to a regular polygon with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003es\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24.5px 8px; transform-origin: 24.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e sides). \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = sumRecipPolyNum(s)\r\n y = sum(1/s);\r\n","test_suite":"%%\r\nn = 3;\r\ny = sumRecipPolyNum(n);\r\ny_correct = 2;\r\nassert(abs(y-y_correct)\u003c1e-13)\r\n\r\n%%\r\nn = 4;\r\ny = sumRecipPolyNum(n);\r\ny_correct = 1.644934066848226;\r\nassert(abs(y-y_correct)\u003c1e-13)\r\n\r\n%%\r\nn = 7;\r\ny = sumRecipPolyNum(n);\r\ny_correct = 1.32277925312239;\r\nassert(abs(y-y_correct)\u003c1e-13)\r\n\r\n%%\r\nn = 11;\r\ny = sumRecipPolyNum(n);\r\ny_correct = 1.19543411652963;\r\nassert(abs(y-y_correct)\u003c1e-13)\r\n\r\n%%\r\nn = 18;\r\ny = sumRecipPolyNum(n);\r\ny_correct = 1.11589671405633;\r\nassert(abs(y-y_correct)\u003c1e-13)\r\n\r\n%%\r\nn = 24:26;\r\ny = arrayfun(@sumRecipPolyNum,n);\r\ns = sum(y);\r\ns_correct = 3.247536806913290;\r\nfor k = 1:3\r\n str = num2str(y(k),'%1.15f');\r\n z(k) = str2num(flip(str(11:14)));\r\nend\r\nfs = sum(factor(sum(z)));\r\nfs_correct = 185;\r\nassert(abs(s-s_correct)\u003c1e-13)\r\nassert(isequal(fs,fs_correct))\r\n\r\n%%\r\nn = 31;\r\nindx = [4 5 7 9 10 11];\r\ns = num2str(sumRecipPolyNum(n),'%1.15f');\r\nd = num2str(sumRecipPolyNum(str2num(s(indx(randi(6))))),'%1.15f')-'0';\r\np = prod(d(3:13));\r\np_correct = 186624;\r\nassert(isequal(p,p_correct))\r\n\r\n%%\r\nn = 57;\r\ns = num2str(sumRecipPolyNum(n),'%1.15f');\r\nindx = [3 4; 4 10; 6 7; 7 10; 8 9; 8 10];\r\nfor k = size(indx,1):-1:1\r\n a(k) = str2num(s(indx(k,1):indx(k,2)));\r\nend\r\nassert(all(isprime(a)))","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2024-07-16T01:35:12.000Z","deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2024-07-16T01:34:17.000Z","updated_at":"2025-10-01T15:44:42.000Z","published_at":"2024-07-16T01:35:12.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAs explained in \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/60571\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eCody Problem 60571\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, a \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Polygonal_number\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003epolygonal number\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e is the number of dots arranged in the shape of a regular polygon. For example, 15 is a triangular number because dots can be arranged in the shape of a triangle with rows of 1, 2, 3, 4, and 5 dots. The number 16 is a square number because dots can be arranged in four rows of four. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to sum the reciprocals of polygonal numbers. In particular, compute\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y = sum[1/P_n,s,{n,1,inf}]\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey = \\\\sum_{n=1}^\\\\infty \\\\frac{1}{P_{n,s}}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"P_n,s\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eP_{n,s}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eth \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"s\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e-gonal number (i.e., the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eth number corresponding to a regular polygon with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"s\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e sides). \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":49743,"title":"Determine aquifer properties: unsteady pump test in a confined aquifer","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 714.15px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 357.075px; transform-origin: 407px 357.075px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 382.358px 7.79167px; transform-origin: 382.358px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAn important task in characterizing the flow of groundwater is to determine the properties of the aquifer, or the underground water-bearing formation. One approach is to disturb the aquifer, observe its response, and fit a theoretical formula to the observations. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 106.633px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 53.3167px; text-align: left; transform-origin: 384px 53.3167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 297.567px 7.79167px; transform-origin: 297.567px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor example, suppose a confined aquifer initially has no flow. In that case, the piezometric head \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eh\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 67.675px 7.79167px; transform-origin: 67.675px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, or the level to which water would rise in an observation well, would be a uniform value \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"h0\" style=\"width: 15.5px; height: 20px;\" width=\"15.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 124.467px 7.79167px; transform-origin: 124.467px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. A well turned on and pumped at a rate \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"Q0\" style=\"width: 19px; height: 20px;\" width=\"19\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 38.9083px 7.79167px; transform-origin: 38.9083px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e will create a cone of depression; that is, it will draw down the piezometric head to a level \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"h(r)\" style=\"width: 29px; height: 18.5px;\" width=\"29\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24.8917px 7.79167px; transform-origin: 24.8917px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003er\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 95.2917px 7.79167px; transform-origin: 95.2917px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the radial distance from the well. Applying conservation of mass and Darcy’s law to this situation leads to a diffusion equation whose solution for the drawdown \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"s = h0 - h\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 79.3417px 7.79167px; transform-origin: 79.3417px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e as a function of distance \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003er\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 30.725px 7.79167px; transform-origin: 30.725px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and time \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003et\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7px 7.79167px; transform-origin: 7px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 44px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22px; text-align: left; transform-origin: 384px 22px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"s = (Q0/(4 pi T)) integral(exp(-x)/x, u, infinity)\" style=\"width: 117px; height: 44px;\" width=\"117\" height=\"44\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 7.79167px; transform-origin: 21.0083px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eT\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 65.9833px 7.79167px; transform-origin: 65.9833px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the transmissivity, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eS\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 67.15px 7.79167px; transform-origin: 67.15px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the storativity, and \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 36.05px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.025px; text-align: left; transform-origin: 384px 18.025px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"u = Sr^2/(4Tt)\" style=\"width: 49.5px; height: 36px;\" width=\"49.5\" height=\"36\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 349.917px 7.79167px; transform-origin: 349.917px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that achieves the objective of a pumping test: to determine the transmissivity and storativity from measurements of drawdown in time. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 347.467px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 173.733px; text-align: left; transform-origin: 384px 173.733px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" style=\"vertical-align: baseline;width: 497px;height: 342px\" 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data-image-state=\"image-loaded\" width=\"497\" height=\"342\"\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [T,S] = confinedPumpTest(t,s,Q0,r)\r\n % t = time, s = drawdown, Q0 = pumping rate, r = distance from well\r\n T = f1(t,s,Q0,r);\r\n S = f2(t,s,Q0,r);\r\nend","test_suite":"%%\r\nQ0 = 0.15; % Pumping rate (m3/s)\r\nr = 100; % Distance from well (m)\r\nt = [100 1000 10000 1e5 2e5]; % Time (s)\r\ns = [0.0402 1.236 3.664 6.276 7.05]; % Drawdown (m)\r\n[T,S] = confinedPumpTest(t,s,Q0,r);\r\nT_correct = 1.042e-2; % Transmissivity (m2/s)\r\nS_correct = 9.864e-4; % Storativity\r\nassert(abs(T-T_correct)/T_correct \u003c 1e-3 \u0026\u0026 abs(S-S_correct)/S_correct \u003c 1e-3)\r\n\r\n%%\r\nQ0 = 0.30; % Pumping rate (m3/s)\r\nr = 100; % Distance from well (m)\r\nt = [100 1000 10000 1e5 2e5]; % Time (s)\r\ns = [0.0804 2.472 7.328 12.552 14.1]; % Drawdown (m)\r\n[T,S] = confinedPumpTest(t,s,Q0,r);\r\nT_correct = 1.042e-2; % Transmissivity (m2/s)\r\nS_correct = 9.864e-4; % Storativity\r\nassert(abs(T-T_correct)/T_correct \u003c 1e-3 \u0026\u0026 abs(S-S_correct)/S_correct \u003c 1e-3)\r\n\r\n%%\r\nQ0 = 0.1; % Pumping rate (m3/s)\r\nr = 40; % Distance from well (m)\r\nt = [300 2000 8000 12000 24000 40000]; % Time (s)\r\ns = [0.494 1.749 2.830 3.153 3.709 4.120]; % Drawdown (m)\r\n[T,S] = confinedPumpTest(t,s,Q0,r);\r\nT_correct = 9.838e-3; % Transmissivity (m2/s)\r\nS_correct = 3.4e-3; % Storativity\r\nassert(abs(T-T_correct)/T_correct \u003c 1e-3 \u0026\u0026 abs(S-S_correct)/S_correct \u003c 1e-3)\r\n\r\n%%\r\nQ0 = 0.1; % Pumping rate (m3/s)\r\nr = 65; % Distance from well (m)\r\nt = [300 2000 8000 12000 24000 40000]; % Time (s)\r\ns = [0.125 1.050 2.067 2.383 2.931 3.339]; % Drawdown (m)\r\n[T,S] = confinedPumpTest(t,s,Q0,r);\r\nT_correct = 9.838e-3; % Transmissivity (m2/s)\r\nS_correct = 3.4e-3; % Storativity\r\nassert(abs(T-T_correct)/T_correct \u003c 2e-3 \u0026\u0026 abs(S-S_correct)/S_correct \u003c 1e-3)\r\n\r\n%%\r\nQ0 = 0.05; % Pumping rate (m3/s)\r\nr = 5+10*rand; % Distance from well (m)\r\nt = [4e5 9e5 14e5 19e5 24e5]; % Time (s)\r\ns = [0.859 0.918 0.951 0.973 0.991]; % Drawdown (m)\r\n[T,S] = confinedPumpTest(t,s,Q0,r);\r\nlogfit = polyfit(log(t),s,1); \r\nTapprox = Q0/(4*pi*logfit(1)); \r\nSapprox = 2.25*Tapprox*exp(-logfit(2)/logfit(1))/r^2; \r\nassert(abs(T-Tapprox)/Tapprox \u003c 1e-3 \u0026\u0026 abs(S-Sapprox)/Sapprox \u003c 2e-3)","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":4,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-01-04T00:26:53.000Z","updated_at":"2026-01-09T18:01:40.000Z","published_at":"2021-01-04T05:23:32.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAn important task in characterizing the flow of groundwater is to determine the properties of the aquifer, or the underground water-bearing formation. One approach is to disturb the aquifer, observe its response, and fit a theoretical formula to the observations. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example, suppose a confined aquifer initially has no flow. In that case, the piezometric head \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"h\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eh\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, or the level to which water would rise in an observation well, would be a uniform value \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"h0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eh_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. A well turned on and pumped at a rate \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Q0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eQ_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e will create a cone of depression; that is, it will draw down the piezometric head to a level \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"h(r)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eh(r)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"r\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003er\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the radial distance from the well. Applying conservation of mass and Darcy’s law to this situation leads to a diffusion equation whose solution for the drawdown \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"s = h0 - h\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es = h_0-h\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e as a function of distance \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"r\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003er\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and time \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"t\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003et\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"s = (Q0/(4 pi T)) integral(exp(-x)/x, u, infinity)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es = {Q_0 \\\\over 4 \\\\pi T} \\\\int_u^\\\\infty {e^{-x} \\\\over x} dx\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"T\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eT\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the transmissivity, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"S\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eS\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the storativity, and \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"u = Sr^2/(4Tt)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu = {S r^2 \\\\over 4 T t}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that achieves the objective of a pumping test: to determine the transmissivity and storativity from measurements of drawdown in time. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"342\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"497\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" 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\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":46012,"title":"Find the zeros of the Bessel function of the first kind","description":"\u003chttps://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html Bessel functions\u003e are important in many problems in mathematical physics--especially those with circular symmetry. Examples include the vibrations of a circular membrane and conduction of heat in a cylinder. Solving such problems usually requires finding the zeros of the Bessel functions and their derivatives. Like sine and cosine, the Bessel function of the first kind oscillates--as in the figure below. However, its zeros are not as easily predicted.\r\n\r\nFind the kth zero of the Bessel function of the first kind of order ν (nu) and its derivative. For Bessel functions with ν \u003e 0, skip the zero at z = 0. For the derivative of the Bessel function of order ν = 0, start counting with the zero at z = 0.\r\n\r\n\r\n\u003c\u003chttps://www.mathworks.com/help/examples/matlab/win64/PlotBesselFunctionsOfFirstKindExample_01.png\u003e\u003e","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 569.467px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 284.733px; transform-origin: 407px 284.733px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eBessel functions\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 305.617px 7.8px; transform-origin: 305.617px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are important in many problems in mathematical physics--especially those with circular symmetry. Examples include the vibrations of a circular membrane and conduction of heat in a cylinder. Solving such problems usually requires finding the zeros of the Bessel functions and their derivatives. Like sine and cosine, the Bessel function of the first kind oscillates--as in the figure below. However, its zeros are not as easily predicted.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 27.2333px 7.8px; transform-origin: 27.2333px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFind the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ek\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 165.683px 7.8px; transform-origin: 165.683px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eth zero of the Bessel function of the first kind of order \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eν\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 138.083px 7.8px; transform-origin: 138.083px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and its derivative. For Bessel functions with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eν\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 30.15px 7.8px; transform-origin: 30.15px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u0026gt; 0, skip the zero at \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ez\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 166.667px 7.8px; transform-origin: 166.667px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e = 0. For the derivative of the Bessel function of order \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eν\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 109.1px 7.8px; transform-origin: 109.1px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e = 0, start counting with the zero at \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ez\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.8167px 7.8px; transform-origin: 13.8167px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e = 0.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 425.467px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 212.733px; text-align: center; transform-origin: 384px 212.733px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" style=\"vertical-align: baseline\" 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\" data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [j,jp] = BesselJzero(nu,k)\r\n% k = number of the zero, nu = order of the Bessel function\r\n% j = zero of the Bessel function, jp = zero of the derivative of the Bessel function\r\n j = ...;\r\n jp = ...; \r\nend","test_suite":"%%\r\nnu = 0;\r\nk = 1; \r\nj_correct = 2.404825557695773;\r\nassert(abs(BesselJzero(nu,k)-j_correct)/j_correct \u003c 1e-4)\r\n\r\n%%\r\nnu = 1;\r\nk = 3; \r\nj_correct = 10.17346813506272;\r\nassert(abs(BesselJzero(nu,k)-j_correct)/j_correct \u003c 1e-4)\r\n\r\n%%\r\nnu = 2;\r\nk = 2; \r\nj_correct = 8.41724414039981;\r\njp_correct = 6.70613;\r\n[j,jp] = BesselJzero(nu,k);\r\nassert(abs(j-j_correct)/j_correct \u003c 1e-4)\r\nassert(abs(jp-jp_correct)/jp_correct \u003c 1e-4)\r\n\r\n%% \r\nnu = 0;\r\nj_correct = [2.404826 5.520078 8.653728 11.791534 14.930918];\r\njp_correct = [0 3.831706 7.015587 10.173468 13.323692];\r\nfor k = 1:5\r\n [j(k) jp(k)] = BesselJzero(nu,k);\r\nend\r\nassert(all(abs(j-j_correct)/j_correct \u003c 1e-4))\r\nassert(all(abs(jp-jp_correct)/jp_correct \u003c 1e-4))\r\n\r\n%% \r\nnu = 3;\r\nj_correct = [6.380162 9.761023 13.015201 16.223466 19.409415];\r\njp_correct = [4.20119 8.01524 11.3459 14.5858 17.7887];\r\nfor k = 1:5\r\n [j(k) jp(k)] = BesselJzero(nu,k);\r\nend\r\nassert(all(abs(j-j_correct)/j_correct \u003c 1e-4))\r\nassert(all(abs(jp-jp_correct)/jp_correct \u003c 1e-4))\r\n\r\n%%\r\nnu = 1/3;\r\nk = 5; \r\nj_correct = 15.450649;\r\nassert(abs(BesselJzero(nu,k)-j_correct)/j_correct \u003c 1e-4)\r\n\r\n%%\r\nnu = 1/2;\r\nk = randi(5);\r\nj_correct = k*pi;\r\nassert(abs(BesselJzero(nu,k)-j_correct)/j_correct \u003c 1e-4)","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":4,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-07-01T22:27:27.000Z","updated_at":"2020-07-30T20:57:30.000Z","published_at":"2020-07-02T02:25:18.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eBessel functions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e are important in many problems in mathematical physics--especially those with circular symmetry. Examples include the vibrations of a circular membrane and conduction of heat in a cylinder. Solving such problems usually requires finding the zeros of the Bessel functions and their derivatives. Like sine and cosine, the Bessel function of the first kind oscillates--as in the figure below. However, its zeros are not as easily predicted.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFind the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"k\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eth zero of the Bessel function of the first kind of order \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"nu\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\nu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and its derivative. For Bessel functions with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"nu\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\nu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e \u0026gt; 0, skip the zero at \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"z\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ez\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e = 0. For the derivative of the Bessel function of order \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"nu\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\nu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e = 0, start counting with the zero at \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"z\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ez\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e = 0.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc 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the generalized hypergeometric function","description":"The \u003chttps://en.wikipedia.org/wiki/Generalized_hypergeometric_function generalized hypergeometric function\u003e is defined as \r\n\r\n\u003c\u003chttps://wikimedia.org/api/rest_v1/media/math/render/svg/1622e60ecca4a7a8287805cbc798387110f49c68\u003e\u003e\r\n\r\n \r\n \r\nThe numbers _p_ and _q_ are the numbers of values _a_ and _b_ in the numerator and denominator (respectively), and the Pochhammer symbol (a)_n is defined by\r\n\r\n\u003c\u003chttps://wikimedia.org/api/rest_v1/media/math/render/svg/c560a95c630b385d8bdf14da55e36d1286d8c68f\u003e\u003e\r\n\r\n`\r\n\r\nMany other functions can be expressed in terms of the generalized hypergeometric function. For example, \r\n\r\n\r\n exp(x) = pFq([],[],x)\r\n cos(x) = pFq([],1/2,-x^2/4)\r\n besselj(0,x) = pFq([],1,-x^2/4)\r\n \r\nThe generalized hypergeometric function can be computed with |hypergeom| from the Symbolic Math Toolbox, but it is not available in Cody or basic MATLAB.\r\n\r\nWrite a function to evaluate the generalized hypergeometric function. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 395.9px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 197.95px; transform-origin: 407px 197.95px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12.0667px 7.8px; transform-origin: 12.0667px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Generalized_hypergeometric_function\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003egeneralized hypergeometric function\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 41.2333px 7.8px; transform-origin: 41.2333px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is defined as\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 45px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22.5px; text-align: left; transform-origin: 384px 22.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6px 7.8px; transform-origin: 13.6px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAloAAABaCAYAAABgxPjDAAAWjUlEQVR4nO2dzZHyPBaFTwKzoiYBEiABIiAClrNjOTsSmOoUyGCqCIEcOgUmhU6hZ2FO6aKWZMmWbdmcp8r1fS/QxsiydHT/BAghhBBCCCGEEEIIIYQQQgghhBBCCCGEEEIIIYQQQgghhBBCCCGEEEIIIYQQQgghhBBCCCGEEEKIj2QP4Pj6rxBCCCGEqMAewAPArzmuAHav93cAvgD8vN67oxNkQgghhBCih2904gnoBBQF1dfrtfvr39+v4/f1GYktIYQQQogEZ3SiyXKAs2zdXv89eH/zi84KJoQQQgghItzQWal8LnBiK2S5+kYn0HaB96biC++Cby5uUNwa0Ans84Lff4WsqEIIIVbGA51oCgkmugkvgfeur/fmElo3LDfJ79G1xRIirxXO6O7BkuzQubGXFHtCCCFEEYy/8i0FOwBPuHgs36Jzfb0/B7fX9y3JAd3v/UTL1hntuIl36ETvaekLEUIIIXJgvJW12BzhAt9v5n0rMp4IW7pqc0HYtbkEZ7RzLXNxQFhoL8ke7V2TEEIIEYUuQns80VkPdub9JzrL0gMuS3FK9ojHiC3FA8tb1+aE97w12A+FEEKI5tmhsxw9XscFf2Ov7Ptzxcjc0N5kekIn/j7BmsKM1DkTHnLZobsPitcSQgghBkBrVouxOE+4GmNbpvXfGcuYFUIIIUQPX+iEVouwSv6WoeWuJbetD6/xk7NBhRBCiEE80Z7bkKxBhIyFSRAtQ/dhizFkQgghRLPQbThkAt3DuZQYU3ZCF7z/M/CcNa9vLQwVunO0v6VlQS6EEEI0yRHxQqkpWETVVnK39cBqWqG4ufZWGfL75mx/8sD23bhCCCEawi/TsPQxpNYWJ+ySSZmurhv+Zsk98F6yogY/2K4lhUK3xPo0d/sTFtxthSu6a/pGF8u3w3stOiGEECsnVBPrjm4CqHnc0U2gz8D3+fW4SikVWpzIHghP5Jzoa1qgprSkcHJ+IN4GB3S/5454qYnT6xylW+eUCq0l2p+wr7QUEM/fe0LXNiwA/CllQYQQYtOwkrcVO1PXQtqhm0y+EBZepSUaSoSW3fw6Ntny/Zo1lziZTgGFC4/QvbPtHLKU7PF+jpIyDSVCa6n2J0Osn1PzAxc7xut64nO3cBJCiM3BbXXsMaeb6wgnRIZ8d+7kuYMTlTFrCTMEU9aE3es7S9ycUwotusMokkMCxorpkNXwgPf7X2LVyhVatdp/DK0JLdvu7E8UvVuO6RNCiI/DTtY85s6SO8FNxCWTbO7kyc+lrCkURN8IW4bOcNahEqvPlNlu3MD6B/FrYtX2J+KWItYis/tY5pArtGq0/1haK0PBNrG/lwufOfYGFUIIMRN+tlffhDgVe7jNqXOhFaTP5cjfF5vEKRhiFp0juvagICkRWnNbCeeE9an62mNs+9dgSsuihe7x0ILhALco4PVY8Usx2FIcmRBCiAr47qOpMr/62Bd+L687ZVGxMUixSdwmBqREG60QuULrEwpl9mVV1mz/MbBe1xTs0PUJ+ztswP8O7y5yLhD8vv40r7W4pZQQQogR2GBlm4U4N2eUuU5+kL5Oay0JCSRm7dn4IG6e7VMqtPjdW540WWA0Rs32H0OpJbKEHZwFyyYo0DL1DZd9+wvgX/j7fHHRcHtd55b7jBBCfCx2wvMDdeekJOvshnRpCGtR8QXZzRx0bdGFGbqGUqFFV2PMXXZ5fYYWrzO6e9DyBs0+jCuKxdbVan+65I7o2utg/v+KtBU05WI+vL6LpUjwOi9LYpRada2wPKO7l74lj/3CXs8enWD9wTRZl01BE598pEKIGNwuZGvY7LC+bLZWyNkw2ApI1gp7whXNtO9/Iy4uS4VWKubM1kw6wQkuxjOtJb2f7tGUIK/R/hRa7JN3OMsPX4uJIpYTCXGAE3pf6ETODS5JZIjosb+lyeKjezjVWnqMHQwksoQQuTywzZWnXZHbCWPueK0SOGnH2OM9u/KOd6FMt2nMkkVKhBZdQamMSGYM3sz3fr9eW4vQAvKsijXan6LOtwaFAsstfXsnWqHFfhSyOuVihWOTemIPt7Gk/7A/vMOvbDwm4HD3Ot8WV6lCiPpwzGilLk9NOMnYo2V3FstDTC0GS4QWrSIx6FKj0AKckFhbliLdXlM/C7Re+e1qK6z7sCxHqm/YrEhCwTukT9lyFi0vUP6sqmKqcAcnysaIpL6HQgghfI6YZ4JfgtAWPS0vRO+YrjQAyRVa7BcpqxStOLb/UEisMUvxgmHbKJVAy5PfDxnb5Lc3XeGpfmszbikUxwpem1jS9ELMXmjfPlEUZUMHO/79mky1Qog22OoijVYKP16r1XFyDq9EjtBiXbK+66CBwIoqWhKbnpwTMGNuKuhqtXN9zMoFuDiuFNQaVlSNEbx+qZSWLcFvKZJ9q5QjxinpIZtpCiEEkBeLs1ZCW/Q0Gdz7ghlrU8XF5AitO/Ji92iFsaKB7iqgXUHbx1Sxi3S1+lYmtpl/zy/Im9cpeO3zS8F7QNl9oMhm9mjrz8vbSqrvpu0yPhOD1qwmA9aEEKugtKL5mvA3Lm7dtTVlUlOf0GLWWh8U51Y02H3uLmi7jfuwwf21oOXJtssNYdcgyyrkEBK8tJwdUOY+tBa0UJzWJXCti+Gb3mKl7GvERfBGic9gi1YH4Vjq/nJQ3WKsFl1yvthq/VmqfS+4ryCtFKF5Kdf6QdFgyxhw3nuicXdTJrXbn5anG1ydq5igzr0PbHPf7Ujx1SeyTnA1vVh4lfj1tM5ozLpl47NCF8YAtxqm1b6qwn2waJrtVLF9jkQeU7SpnSxqZPTwGnWfx8OSLmMsELXub+w6eI2xyWPrFcBDW/RsNQkgRKzs0NDn3+55578u70oYip+xbW+JPdd8vQ/7PIQyG3963l8UW9ohVbJ/LDn7ZIXYvf6GIu0KV8236doZDTN1m+Zmsfad447ugbmaa/tCYw/QCtjD1d5hoUauIIcMoGPu7wXvdW/8RZztgzFLw5g97WzF6zHH1KI/tEXP2soQiHXC57u1/sZ5ioVXfU497y+KfZAv6BqZJfIf5vWxMLOgxJdszcd2MLeF1OSKLGOuNqWAL7Vg2hIidiK1W0tssXDlVNDN5m9vwT4wNNZp6P1ln7OB3xxfuFHsDf3jztCJIFQgdMgxx1Y5ofqGS2zRIz4Ljhnqa5XwBx1boNSa4Wqs3njzcmMNDuYaQitmXttWg2KnoPU2tS4p/yEPTcwiDQWLL4ZS+6HNie1vJ5RlMz8xTGgx4HbsMYfbMrRFjyz4YkrsQvwLCteogjXThwatmnsHlQgtKwhC1gtrfZF1I481tCktqKG+SHenRFYefN5isT3MtFrSxM77HarXk/u3W8dfDLe+F6JYN4xnm8tF/hHY7JZQvMM98voQSoQWrysm8uiG/EV6cN5DHYXUatOpsBaroRNJrezYtWMtVi0LU5uSXXqdnyK0gPd22moCgBCbxFowYpNbzfoouULLTrixQYUukZho2GHcJpVbo0abTg3N1UPcQXu4iVdC670WU8tYa03pc8rwhk+AsVqy3guxMuzkGxuwalqDcoUWJ9xYvIZdrYeykugSoZtMQmt8m06NTWcv/f4vvLvAJbRc32+qjkwAe99LLec5dXdCrCXrkLBvLxU3+U8A/9WhY8ZjU9hV7xwBsVy9plZlOROuve6UaEvtMP5J1GzTqbCp7EPvl4RWxxjROifcPoPXWioKh45ba8o65GJ4yTT7f6NOe+nQkXtsKtzHDnJzmKRpNUmtXPsmXH+QTE2qElodNdt0KmrEoCx5/S1h72fLQuuOTuCHts7oI2csibGWrEPex28s36cPOnTMeIT4D1ZoSbOuojkVZF9KdmrCPcAV2OQABHQiMbS67BNax9f3nVE2kPHv+tqs9ucsOzgXaaxjkpptWgJ/V86EZN3YMdF/SLwH9E/WrA1XOkGyLfr6SO3PWfZwGZd9/cQ+2zF3E/vPmAm85P76XOD6WihOi+eOwcSN1relGQqfy1o7cgixdv6HRixpZ7jqzw+4gqM/+LvDurVylNSuiXF4ffcdbqUaWvV9IR3AajPf7Gr8CJfWbGt8nBEvtZ8SWr7QzHVB2L9LpaPX/pyPv/FsSmzVatMjXEHJH7gd05mab9vZb98+0dZX14nXmnoQUkLLCrmc6wn9XWqBUPtzPtb6nLO1xDPx2T26e5gSrayw/AVXadlWkS+9vwc4UXbG375uheEh8L7PlvdMpUtVZRyEcPwDy1jSgtjaOZz4fD9/KDZizMqW57eDLc/trzgP6HcP+dfGiZ0NYYVJatJJCS1eR6matX+Xmvhrf87HrxjdFyhbq02P5nOswbQ3r/Hv/PbNib+xlrcrXNDxDXl9NCW0/O1MchcXffuATvU5H79oZZ+r31qJ7nBteUX321ODCkU8P2Pv+dD7a7fcCS0M/P7cN+hxK6GtYQv2rjHD8IDuXs/p6pyreGzr3DBvnznh724THwMHLCt6OFA+0d0I7sJtj6H1siiy/EEvtfLm98VgaQZ+zu883Cutz/XR5zqkxY8DW67IYQfre7hrf85ygLuPP+gXWrXalELBv38hscr25eo8Bwor9ktWDM8hJbR25rxPlFlxr3CWljk/Z6GFiX06Z0Ddw91ze9/7LEX+M8Pxw7/nJfeX139H+HezP8fe989V8ryuiZyth1plCZFFHlinMK3F3CKLHNFGDOHs0K9vfzgHptqZhXs4t2To+2KTP1fDU5vFc4Phj6jjPl2KOQcZCnlrqWQ9tpiwprCYmpTQssx1PVMxlciIPbdfSIu7uduTYRFbg+28xr7J8IOlxC8tgVuN2Utxw7LWXYaQfAyxledUqyQODP6glyNwuBqeklyhdcc6V5CAG+DmWlGEBBWtmiEhTzf1HLEmOUJrzuuZggumK8VCy64voum2jFkK52xPBtFvbQW9dBmHMQs1ipylx1DGcW6tb6RIxSjPyRoXP4Nr4THGxT40HCinGJwY92MHWa6Kczo8U7unIkdoLb0aGMMBf5McpoRC3l9xs539TssBeK74iT6hNff11IYrxykGVcbZ+YM2F1MhATB3e3JRsVaRHIMW/qUE5AXjRBLj/lqACVmfAOf2FlymHD/W8mxyXBskzEOxRiFXTy38uBxmC+WuzHYIxwvVok9onSf87qlhzNWcpvqQC4kxWyHBPDT1fyh9QosB9muEWXtTTcQ2jpPYDMml7+8OdbcDawVbxmGJ35aT4ZmipckecKJ1rc95CS0JXMDFDK8Bm1xU1HdtnAxhCvRUnY5ChiUBvuEm45IV0hemuUZa3FoZBNaOL+RD2aZLYcsNbDFIemqsi5D1xjhwtrBSLU0eWAM2w3ApkfWNcRYgbnXWEt9YZ5xbCXxeWxh7yZqSVGyGc1Eb8kcy7ZlZCFP+6D1cJhcLKbYwMB/wnm3FzLWl/dhrhkKeGY5fyCuWOgcnvGcqTiXct84RLvOP2YmpJAcxjiXLOJxQZy/YJ9pz1TGEZst9lovcFsZfS6gCQYvw2Svu/wx4X9J6w4G59Y1sRTl8sLe+UhQOLt7WMHCuDY7XcwcQs8wGJ5kx1ii66cZYVewG37WgG3ytsZg53DHeknhA/c3RWS6pdewzUCRW6SZbUsXHamqJ9ROqryS2je75NNDiMseihdtWcceQX+8YM1YzPrNUJO3g4ouYEPX9+ncNwcUF/9qy4Er4wbB4KFqtn+jah6E+X6ijHXi+1uHYVrTFlc0YWpLccgpifXCQXoP/XYyHMSDaa68ufoJBrcO6zllE1hdVoWOM64mTakn/oMvSr7nFia9W5uWSpTKmhkKyRKgz8cwXoDa2tYY3jIuIOUM3/O+ilTTVt+0OOlnYOKmlalmEqnkrRmY72Hi3uTMdxfz4u0fontchtO3XksdYFw8X1rlQZIZi/igMawXWc7/fLULXaO5cb5MufDevFf41Aut5vqnmf2Zf0yr3CxcjyEx828dj4pHW2DW4OYUQQmTi71e59DF2Yi0RWnbvzdAkfELd5JpSEVgC67mlxA43Trd7DvtQFJRW1C8VWmyLWC28K+plL5ZeWym0Vtm9UZm4wwzak3kvJqRixd2FEEKIZigRM771oZQdykTYlELL3wA9JJKsqA6F9PjWzRI3YImYsRarOTIUpxZaxFqugL87E/S166HnfSGEEGJxcsWMneyHuJQu6IRLSeD+A9PV92I8GV2dIaFlY+RCVhUbG/WLst9WImZ4HXO5UXltU9f3oiv0DlfOilgRm4oXn+M6hRBCfAi1SygA+ULLWoBKAt2P6CZRTqolYmRoVl4OO7itfmIT+eH1fqrQLkttlGb85QotKzjmqgQwNBO1BCYDUED6FkMr7FOkYriEEEKILGgNYgxUrRIKgMvc6osvohttaOAxv6dELGw9/ibH7UXR02fZqQnv1ZRuSiukQv2Pwr5PaG9xSy8hhBAzwonWTrLcMqfGBJObYTY2y7FUaNEtt+U6WjlZlWy3OYUW3apzfEesT1DY97kFH1A2tRBCiIHYOmgWipAa1p5cQcM4oVSNLG7hFqJUaDHrLCUA/eD62hXSpyZH0FjLT8xFdki8N4Rv9AvAA1w/2KPcwkohFYrBs+7SvsWErFlCCCEGw0k2NOlxoqohLHI2pbYWCH9yY7xTyg1WKrRuiAfC031KgciNtTlxr2V/RIrJlFiwAfehe3REuiAx61Ixjoy19WLCmt8XsiSx1qZt5y+4vphrfeyLO2NfWbp4uxBCiIaYwrpit1bxqbmDB92TKYFC6xqtWid0v/GMbkLss6iUCq1nz2f5+7k10c68tiarVk4mpnUfXuGSIm5IWxgPr/PfzGfYRqnaYam+YGtXsag5X8sts2B/T0hkUjTzfBeEhd/u9bqsWkIIsWFyrSvHwsMKh5TQqpGJRhHVFw/Dic1uE5RyF1pKhBYtPanz0uJhY3S4LdBaLFqAi7frI7RzS0pkU2Q98N4e7DcxcfKNPMuktYpRqOeWWeA1hCxW1oJ3Qif8YnGBgzeVFkIIsS5yrCu/hcecQgvoJskpRUqJ0Orbhs5WDafYWGvxyh26dq9dC4pi34pVllSIueTohkz1AZ7XCjhmCOYIblvWIdQXrFuRm5THrscWlFWJByGE2DBTWVfmFFo851RZfrlC64L+TanpUrWxa7SqrHHCzRE4JVCI+rF9qXtAwZdqPyuSbPA7+3oOBzirXMwKxZiyK9JtMrSGmRBCiBUxpXWFgccpoVVTGHGvuSkKVeYILe4/2OcGolXFXietKmudcM+oV5yV/cZayfZwFqDQ/e1LZgDCAo59fa5iqkIIIT6MXOvKkBgtnic0iYXERg1ocagd89IntOgq7fvemPvrB12b7LHe2ltX1HF9Wlc24AR0LCuTLu8+2Ndtv7Z13q5YVyKCEEKIFZBrXRkSo0VrQSgYmJ+bgh2631DTOkQrS2xCT21zY6FVxQo2mwl3x7on+zPGx2vZgPUT3GbNoRIRFEg5sK+HguuvBecRQgghspjDuhIqWRASG62yx3um4h2dmBgqhnguK8ooDO9Q9hnQtccVLn7pACe+hoo4Whz9PkeX5wXrddsKIYRolDmsK4ytYTYj3UBDt8IRn0nIGiWEEEI0zVzWFVoovl7HGjPrxHKwNpXEuRBCCCFEZUqr8gshhBBCiB72cDWpbHycEEIIIYQQQgghhBBCCCGEEEIIIYQQQgghhBBCCCFa5f8Jmn+w4Uk8SQAAAABJRU5ErkJggg==\" alt=\"(See the definition at the Wikipedia link)\" style=\"width: 301px; height: 45px;\" width=\"301\" height=\"45\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.8167px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.4167px; text-align: left; transform-origin: 384px 21.4167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 41.2333px 7.8px; transform-origin: 41.2333px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe numbers\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ep\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6167px 7.8px; transform-origin: 13.6167px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eq\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 82.85px 7.8px; transform-origin: 82.85px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are the numbers of values\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6167px 7.8px; transform-origin: 13.6167px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eb\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 180.1px 7.8px; transform-origin: 180.1px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in the numerator and denominator (respectively), and the Pochhammer symbol \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADYAAAAoCAYAAAC1mQk2AAACHUlEQVRoge2YaZGEMBCFPw84iAEMoAAFOMABDsYCGkYCHrAwGrDA/ki6yGYrF0OY3a28KmoOQqdf3wQqKip+G7oLZSmguVDeaczAcKG8FljM58dwNSmBAl58iNwDeBaU36LJ3RqWPbDdsGlp4/3AC5hu2KdBG7C/YS8G7vGWYEYbsjheZrO70AI717YU7yYlKmEIG4WNOaGJqZPPd0bGxNGIe/M7FNoLhcNxQRPLRY9WbDXfO/N9NfJiSotBi+W1KJeDGa2UW7Y78/9OPMyEWLE829FeS4WPFHwnFivnfeK608gh9jDrfa1hiNy3IUYo1jtTidneGD1rxJs58j5OTIrM5rkvE0WqskKsWJvZiBOTXhdSerLWpBSE4sVjwe8FVwmfIra3YrJcmWf7ZxRSEGLNdA+se3IQSy1E8kwxSNkNhUSI2GwuO78adG8MyXzhf31RHBMNRt5orqyGvqE958PIQUzWKaPYk+85OKJJ+SonhOfTgcOQItueaLL6Xuw1ojH3d+uyjdE690KkQHsiFIaSHhNHaEtOZlVRRdwajRE6mU83JAY0oZRiEHupFe+s1j7ixezZciJ/ZjyDkXB0iJFd7wjRU4jlxrtQ6BAMnVTJWGaTl2YeqgNBSC6VOiJbieeIjGX2ukuauaLM4WbqeaUUKTuXVuu/t9Kl4drhVF5EY5D8cpu8hKZdTP4UWrRB3WhJOWqoqKioqPi/+ALBAL8AwX/ymwAAAABJRU5ErkJggg==\" alt=\"(a)_n\" style=\"width: 27px; height: 20px;\" width=\"27\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 41.2333px 7.8px; transform-origin: 41.2333px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is defined by\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21.8167px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.9167px; text-align: left; transform-origin: 384px 10.9167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6px 7.8px; transform-origin: 13.6px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"(a)_n = a*(a+1)*(a+2)...(a+n-1)\" style=\"width: 184px; height: 20px;\" width=\"184\" height=\"20\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 327.917px 7.8px; transform-origin: 327.917px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eMany other functions can be expressed in terms of the generalized hypergeometric function. For example,\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21.8167px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.9167px; text-align: left; transform-origin: 384px 10.9167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6px 7.8px; transform-origin: 13.6px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"exp(x) = pFq([],[],x)\" style=\"width: 104.5px; height: 21px;\" width=\"104.5\" height=\"21\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 39px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 19.5px; text-align: left; transform-origin: 384px 19.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6px 7.8px; transform-origin: 13.6px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"cos(x0 = pFq([],1/2,-x^2/4)\" style=\"width: 160.5px; height: 39px;\" width=\"160.5\" height=\"39\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 39px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 19.5px; text-align: left; transform-origin: 384px 19.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6px 7.8px; transform-origin: 13.6px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"besselj(0,x) = pFq([],1,-x^2/4)\" style=\"width: 149px; height: 39px;\" width=\"149\" height=\"39\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.45px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.2333px; text-align: left; transform-origin: 384px 21.2333px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 379.9px 7.8px; transform-origin: 379.9px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere the dash means that the list of parameters is empty. The generalized hypergeometric function can be computed with\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 34.65px 7.8px; transform-origin: 34.65px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 34.65px 8.25px; transform-origin: 34.65px 8.25px; \"\u003ehypergeom\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 252.3px 7.8px; transform-origin: 252.3px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e from the Symbolic Math Toolbox, but it is not available in Cody or basic MATLAB.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 213.05px 7.8px; transform-origin: 213.05px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to evaluate the generalized hypergeometric function.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = pFq(a,b,z)\r\n y = f(a,b,z);\r\nend","test_suite":"%% exp(x)\r\na = [];\r\nb = [];\r\nz = 1;\r\npFq_correct = exp(z);\r\nassert(abs(pFq(a,b,z)-pFq_correct)/pFq_correct \u003c 1e-8)\r\n\r\n%% cos(x)\r\na = [];\r\nb = 1/2;\r\nx = pi/4;\r\nz = -x^2/4;\r\npFq_correct = 1/sqrt(2);\r\nassert(abs(pFq(a,b,z)-pFq_correct)/pFq_correct \u003c 1e-8)\r\n\r\n%% J_0(x)\r\na = [];\r\nb = 1;\r\nx = 1;\r\nz = -x^2/4;\r\npFq_correct = besselj(0,x);\r\nassert(abs(pFq(a,b,z)-pFq_correct)/pFq_correct \u003c 1e-8)\r\n\r\n%% 1/(1-x)^a\r\na = 2;\r\nb = [];\r\nz = 1/2;\r\npFq_correct = 4;\r\nassert(abs(pFq(a,b,z)-pFq_correct)/pFq_correct \u003c 1e-8)\r\n\r\n%% Example from \"help hypergeom\"--current version of help gives hypergeom(1,2,3) = exp(1)-1\r\na = 1;\r\nb = 2;\r\nz = 3;\r\npFq_correct = (exp(3)-1)/3;\r\nassert(abs(pFq(a,b,z)-pFq_correct)/pFq_correct \u003c 1e-8)\r\n\r\n%% Hypergeometric function F(a,b; c; x)\r\na = [1 2];\r\nb = 4;\r\nz = 0.2;\r\npFq_correct = 1.113869211474147;\r\nassert(abs(pFq(a,b,z)-pFq_correct)/pFq_correct \u003c 1e-8)\r\n\r\n%% \r\na = [1 1];\r\nb = 2;\r\nz = rand;\r\npFq_correct = -log(1-z)/z;\r\nassert(abs(pFq(a,b,z)-pFq_correct)/pFq_correct \u003c 1e-8)\r\n\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":6,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-07-11T20:25:32.000Z","updated_at":"2026-01-09T17:23:16.000Z","published_at":"2020-07-11T22:27:35.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Generalized_hypergeometric_function\\\"\u003e\u003cw:r\u003e\u003cw:t\u003egeneralized hypergeometric function\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e is defined as\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"(See the definition at the Wikipedia link)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e_pF_q(a_1,\\\\ldots,a_p; c_1,\\\\ldots,c_q; x) = \\\\sum_{n=0}^\\\\infty \\\\frac{(a_1)_n\\\\cdots (a_p)_n}{(c_1)_n\\\\cdots (c_q)_n}\\\\frac{x^n}{n!}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe numbers\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"q\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eq\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e are the numbers of values\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"b\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eb\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e in the numerator and denominator (respectively), and the Pochhammer symbol \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"(a)_n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e(a)_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is defined by\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"(a)_n = a*(a+1)*(a+2)...(a+n-1)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e(a)_n = a(a+1)\\\\cdots (a+n-1)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eMany other functions can be expressed in terms of the generalized hypergeometric function. For example,\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"exp(x) = pFq([],[],x)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ee^x = {}_0F_0(-,-,x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"cos(x0 = pFq([],1/2,-x^2/4)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\cos(x) = {}_0F_1\\\\left(-,\\\\frac{1}{2},-\\\\frac{x^2}{4}\\\\right)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"besselj(0,x) = pFq([],1,-x^2/4)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eJ_0(x) = {}_0F_1\\\\left(-,1,-\\\\frac{x^2}{4}\\\\right)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere the dash means that the list of parameters is empty. The generalized hypergeometric function can be computed with\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ehypergeom\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e from the Symbolic Math Toolbox, but it is not available in Cody or basic MATLAB.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to evaluate the generalized hypergeometric function.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":46066,"title":"Evaluate the logarithmic integral","description":"The \u003chttps://en.wikipedia.org/wiki/Logarithmic_integral_function logarithmic integral\u003e li(x) plays a role in number theory because the related function Li(x) = li(x) - li(2) provides an estimate for the number of primes less than x. MATLAB's Symbolic Toolbox has the function \u003chttps://www.mathworks.com/help/symbolic/sym.logint.html |logint|\u003e, but it is not available in basic MATLAB or Cody. \r\n\r\nWrite a function to evaluate the logarithmic integral.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 123px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 61.5px; transform-origin: 407px 61.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12.0667px 7.91667px; transform-origin: 12.0667px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Logarithmic_integral_function\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003elogarithmic integral\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"li(x)\" style=\"width: 30.5px; height: 19px;\" width=\"30.5\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 184px 7.91667px; transform-origin: 184px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e plays a role in number theory because the related function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPwAAAAmCAYAAADgDvYuAAAFLElEQVR4nO2dXbXrIBCFt4c4iIEYiIIqqIM6qINaiIZKqIdaqIZaOOeBzuq0DTDAQH7WfGvlpbmHJDtsmAHCBQzDMAzDMAzDMAzDMAzDMIQcX4cWE4Besby9YXrX5wJgUCqrf5XXlRRyBHCOHDllnhJv7AhXYTTpAdyhJ/gSDHBa3gFcPf/G9C5jhDPSA07Hb7rX76mN4yT4mxFvn50Q126E0zjb9AOAw6uQP3Y8X7+PieUdWRmXhL+5JV5HygD3IrfW8/RwBn/greec4U3vfEY4HZ54azhn+ImdPwjLnhDuLEd8vlt+XBE29BHOr0X0XxeVPtjczaRUwAFO8JoVREWgheDvRcPwpvcvXEMNw1NE5mNk5d3w2+jQ7yHTT5A38F74BUvCMsoNJWHHA3lpQyq3RtepQcjwgOldCjdgKKSXmJ0aaF9k3MG9gxt+G90j4tEGL+cZuI4ILcNLoQcsGoQQcoB7ri2Emt/EDC/F9J4nZvgUJoTTpSOc4X3vgEcbsbSLxh2yaW34BxTCkhVfTwstw5ve82gZnnr3UCRwi5wH3r38U+F6QVoannqAopAkkQviIq4RDcOb3n60DH95lRFCMptyg8zwQHj2JopmDn+N3AgNhKTQwVXcM34rLuVZobxxiUqvgSSHN73zkebwd895gnLzUsjwEiNLGhkvJYbv4SoVH3QI3XCKODRFxUczeR404PO6vrlPCoFyBpN6xNcrSI6cyj+n5971bonP8COcDn+e8xzNZyVtJaE65fxZjapGDz9AVgFzQ9QzK3/Ee9STepu50U+N6/JKUXLkhIwhPfeqd0tiPfwpcp6XUTroR+9Turim6LpaIX2sAtJN5rSGvIKf4CpcysM+kRd2afXwObrG9Nyj3i2JGV6S41PDWJq+UOolLYfeT1ZksQXDA5+jmKnLQylM3RJLGh5orzdfbtoidVqL4cm8qTMb2VHUVgzP86rUeV4aENkSSxu+td48jWiROq3B8B3yR9z/kBlFbcXwPKdKXURihk+ntd4aPXxK/V2D4WnBTs6iqN0bnq9tTv2CKXfqZG2j9Cnnt6h3S5Y2fInZi2YHcgzf4feFxipgh7xcBXgvTaS8MjWnzG0N1zpKLzm/Rb1bomF4WnOQuuqtxOz83poZ/ozfL4NiFRDIG72lTy/5/GjKWmKq+EvOwy8xSg9sT++WaBg+Z7ScpjVDZj8gXGeKFjelGp4WYHw/pKQCXpE2ekuDGhRS8rySBpKuCLew9OKy1x4vhIbhTW8/GoYHnL7SQTf6kCk0CEr+CjUIlEokRwip38PzT/m+WxdJBaQVQqEHvuA973vHZ0jK54cvcKFRLNykZYgtvhbTgnrJUsOb3n6olyw1/ARZBETvgsY3fIckhbojY++BgV2AH6EboYO3QB0+p1Ro15w5qCKHBOTXmavMsd1gvrlDf1unmtDyWZ+epnc5Az53e/reoouWGfvOc6jhCEXH/PNXyREK1emdJg2kSva0Cx1UwUI5rq+SxVpEmp4JrdUOnedQD7X2DzmImJ6mdzmh6b+D4PwcD4QbuRRvxQZ4T2i3v4EKPRR27RAyYf1rumtjetfngHYm9G26uWpOKNy1Q8CI+vu4bQXTuz5X1E9l5mbHNsOEejuj0Nddax8pbonpXRea4ailQYuNSKtzg+5/ikBcK5W7dUzvutTao58ak63t/T/LBN3KclEub2+Y3nWhVaha5uzgGtRdmJ3QHOzYdMjTCNO7Pload4plGYZhGIZhGIZhGIZhGIaxFP/Ka2gIYQHNwgAAAABJRU5ErkJggg==\" alt=\"Li(x) = li(x) - li(2)\" style=\"width: 126px; height: 19px;\" width=\"126\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.0667px 7.91667px; transform-origin: 40.0667px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e provides an estimate for the number of primes less than x. MATLAB's Symbolic Toolbox has the function\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/help/symbolic/sym.logint.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003elogint\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 75px 7.91667px; transform-origin: 75px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, but it is not available in basic MATLAB or Cody.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 158.967px 7.91667px; transform-origin: 158.967px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to evaluate the logarithmic integral.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 29.1833px 7.91667px; transform-origin: 29.1833px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSee also \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46081-set-soldner-s-constant\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problem 46081 \"Set Soldner's constant\"\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.91667px; transform-origin: 1.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = logarithmicIntegral(x)\r\n y = f(x);\r\nend","test_suite":"%%\r\nx = 0;\r\ny_correct = 0;\r\nassert(abs(logarithmicIntegral(x)-y_correct)\u003c1e-8)\r\n\r\n%%\r\nx = 0.2;\r\ny_correct = -0.085126486728794;\r\nassert(abs(logarithmicIntegral(x)-y_correct)\u003c1e-8)\r\n\r\n%%\r\nx = 0.5;\r\ny_correct = -0.378671043061088;\r\nassert(abs(logarithmicIntegral(x)-y_correct)\u003c1e-8)\r\n\r\n%%\r\nx = 1;\r\ny = logarithmicIntegral(x);\r\nassert(isinf(y) \u0026\u0026 sign(y) == -1)\r\n\r\n%%\r\nx = 2;\r\ny_correct = 1.045163780117493;\r\nassert(abs(logarithmicIntegral(x)-y_correct)\u003c1e-8)\r\n\r\n%%\r\nx = 5;\r\ny_correct = 3.634588310032652;\r\nassert(abs(logarithmicIntegral(x)-y_correct)\u003c1e-8)\r\n\r\n%%\r\nx = 8;\r\ny_correct = 5.253718299558931;\r\nassert(abs(logarithmicIntegral(x)-y_correct)\u003c1e-8)\r\n\r\n%%\r\nx = 3+2i;\r\ny_correct = 2.558790740400258 + 1.594445119241119i;\r\ny = logarithmicIntegral(x);\r\nassert(abs(real(y)-real(y_correct))\u003c1e-8 \u0026\u0026 abs(imag(y)-imag(y_correct))\u003c1e-8)\r\n\r\n%%\r\nx = 3-0.2i;\r\ny_correct = 2.169086896211800 - 0.181703645882027i;\r\ny = logarithmicIntegral(x);\r\nassert(abs(real(y)-real(y_correct))\u003c1e-8 \u0026\u0026 abs(imag(y)-imag(y_correct))\u003c1e-8)\r\n\r\n%%\r\nx = -0.3-2i;\r\ny_correct = 0.999726888286245 - 3.238096925989443i;\r\ny = logarithmicIntegral(x);\r\nassert(abs(real(y)-real(y_correct))\u003c1e-8 \u0026\u0026 abs(imag(y)-imag(y_correct))\u003c1e-8)\r\n\r\n%%\r\nx = -5-3i;\r\ny_correct = 0.500720609942772 - 5.020802115037742i;\r\ny = logarithmicIntegral(x);\r\nassert(abs(real(y)-real(y_correct))\u003c1e-8 \u0026\u0026 abs(imag(y)-imag(y_correct))\u003c1e-8)\r\n\r\n%%\r\nx = 0.3i;\r\ny_correct = 0.074754684076440 + 3.053886180297906i;\r\ny = logarithmicIntegral(x);\r\nassert(abs(real(y)-real(y_correct))\u003c1e-8 \u0026\u0026 abs(imag(y)-imag(y_correct))\u003c1e-8)\r\n \r\n%%\r\nx = 0.4-2i;\r\ny_correct = 1.171012435933119 - 2.861745062394908i;\r\ny = logarithmicIntegral(x);\r\nassert(abs(real(y)-real(y_correct))\u003c1e-8 \u0026\u0026 abs(imag(y)-imag(y_correct))\u003c1e-8)","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":16,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-07-28T15:11:20.000Z","updated_at":"2026-01-29T16:19:49.000Z","published_at":"2020-07-29T01:46:40.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Logarithmic_integral_function\\\"\u003e\u003cw:r\u003e\u003cw:t\u003elogarithmic integral\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"li(x)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e{\\\\rm li}(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e plays a role in number theory because the related function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Li(x) = li(x) - li(2)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e{\\\\rm Li}(x) = {\\\\rm li}(x) - {\\\\rm li}(2)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e provides an estimate for the number of primes less than x. MATLAB's Symbolic Toolbox has the function\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/help/symbolic/sym.logint.html\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003elogint\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, but it is not available in basic MATLAB or Cody.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to evaluate the logarithmic integral.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSee also \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46081-set-soldner-s-constant\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem 46081 \\\"Set Soldner's constant\\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":54610,"title":"Evaluate the Kelvin functions","description":"The Kelvin functions ber, bei, ker, and kei are related to Bessel functions of order . When the order is not specified, the default is . The functions ker() and kei() appear in the solution for velocity in the boundary layer under water waves. \r\nWrite a function to compute the four Kelvin functions. Allow for a variable number of outputs. If the order is not specified, take it to be zero. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 114.75px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 57.375px; transform-origin: 407px 57.375px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63.75px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.875px; text-align: left; transform-origin: 384px 31.875px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 75.075px 7.50833px; transform-origin: 75.075px 7.50833px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe Kelvin functions ber\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"_nu(x)\" style=\"width: 27px; height: 20px;\" width=\"27\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.225px 7.50833px; transform-origin: 13.225px 7.50833px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, bei\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"_nu(x)\" style=\"width: 27px; height: 20px;\" width=\"27\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6083px 7.50833px; transform-origin: 13.6083px 7.50833px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, ker\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"_nu(x)\" style=\"width: 27px; height: 20px;\" width=\"27\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 26.45px 7.50833px; transform-origin: 26.45px 7.50833px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and kei\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"_nu(x)\" style=\"width: 27px; height: 20px;\" width=\"27\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 124.075px 7.50833px; transform-origin: 124.075px 7.50833px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are related to Bessel functions of order \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eν\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 72.725px 7.50833px; transform-origin: 72.725px 7.50833px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. When the order is not specified, the default is \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"nu = 0\" style=\"width: 36px; height: 18px;\" width=\"36\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 59.8917px 7.50833px; transform-origin: 59.8917px 7.50833px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The functions ker(\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 29.175px 7.50833px; transform-origin: 29.175px 7.50833px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e) and kei(\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 175.825px 7.50833px; transform-origin: 175.825px 7.50833px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e) appear in the solution for velocity in the boundary layer under water waves. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 372.867px 7.50833px; transform-origin: 372.867px 7.50833px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to compute the four Kelvin functions. Allow for a variable number of outputs. If the order is not specified, take it to be zero. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [ber,bei,ker,kei] = kelvin(x,nu)\r\n ber = besselr(nu,x);\r\n bei = besseli(nu,x);\r\n ker = kesselr(nu,x);\r\n kei = kesseli(nu,x);\r\nend","test_suite":"%%\r\nx = 3;\r\ntol = 1e-15;\r\n[ber,bei,ker,kei] = kelvin(x);\r\n[ber_correct,bei_correct,ker_correct,kei_correct] = deal(-0.221380249598694,1.937586785266043,-0.067029233303799,-0.051121884045987);\r\nassert(abs(ber-ber_correct)\u003ctol \u0026 abs(bei-bei_correct)\u003ctol \u0026 abs(ker-ker_correct)\u003ctol \u0026 abs(kei-kei_correct)\u003ctol)\r\n\r\n%%\r\nx = 1:4;\r\ntol = 1e-15;\r\nber = kelvin(x);\r\nber_correct = [0.984381781213087 0.7517341827138085 -0.221380249598694 -2.563416557258581];\r\nassert(all(abs(ber-ber_correct)\u003ctol))\r\n\r\n%%\r\nx = -0.4;\r\nnu = 1;\r\ntol = 1e-15;\r\n[ber,bei] = kelvin(x,nu);\r\n[ber_correct,bei_correct] = deal(0.1442308644531633,-0.1385741359112079);\r\nassert(abs(ber-ber_correct)\u003ctol \u0026 abs(bei-bei_correct)\u003ctol)\r\n\r\n%%\r\nx = 3.4;\r\nnu = 0.5;\r\ntol = 1e-15;\r\n[ber,bei,ker,kei] = kelvin(x,nu);\r\n[ber_correct,bei_correct,ker_correct,kei_correct] = deal(-2.245652084214816,0.82701468622679,-0.05553905648843065,0.02619201937598225);\r\nassert(abs(ber-ber_correct)\u003ctol \u0026 abs(bei-bei_correct)\u003ctol \u0026 abs(ker-ker_correct)\u003ctol \u0026 abs(kei-kei_correct)\u003ctol)\r\n\r\n%%\r\nx = [-psi(1) exp(1) pi];\r\nnu = 1/3;\r\ntol = 2e-14;\r\n[ber,bei,ker,kei] = kelvin(x,nu);\r\nber_correct = [0.4900252831480887 -0.7214812981316725 -1.432808723213069];\r\nbei_correct = [0.5553968849316957 1.532550247673432 1.535388618042967];\r\nker_correct = [0.2909569660163571 -0.1038243609935959 -0.07552171571938433];\r\nkei_correct = [-0.983688891425642 -0.03413110664720041 -0.001259453778330345];\r\nassert(all(abs(ber-ber_correct)\u003ctol) \u0026 all(abs(bei-bei_correct)\u003ctol) \u0026 all(abs(ker-ker_correct)\u003ctol) \u0026 all(abs(kei-kei_correct)\u003ctol))\r\n\r\n%%\r\nfiletext = fileread('kelvin.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'regexp') || contains(filetext, 'import'); \r\nassert(~illegal)","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2022-05-06T01:51:35.000Z","deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2022-05-06T01:49:11.000Z","updated_at":"2026-01-09T20:00:53.000Z","published_at":"2022-05-06T01:51:35.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe Kelvin functions ber\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"_nu(x)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e_\\\\nu(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, bei\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"_nu(x)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e_\\\\nu(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, ker\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"_nu(x)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e_\\\\nu(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and kei\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"_nu(x)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e_\\\\nu(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e are related to Bessel functions of order \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"nu\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\nu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. When the order is not specified, the default is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"nu = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\nu = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The functions ker(\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e) and kei(\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e) appear in the solution for velocity in the boundary layer under water waves. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to compute the four Kelvin functions. Allow for a variable number of outputs. If the order is not specified, take it to be zero. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":45997,"title":"Evaluate the zeta function for complex arguments","description":"\u003chttps://www.mathworks.com/matlabcentral/cody/problems/45988 Cody Problem 45988\u003e involved computing the Riemann zeta function for real arguments greater than 1. Code that works for that problem can reveal the connection between pi and the values of the zeta function evaluated at positive even integers; this connection is explored in \u003chttps://www.mathworks.com/matlabcentral/cody/problems/45939-estimate-pi-from-certain-values-of-the-zeta-function Cody Problem 45939\u003e. However, to test the Riemann hypothesis--that all non-trivial zeros of the zeta function have a real part of 1/2, one needs to compute the zeta function for complex arguments.\r\n\r\nWrite a function to evaluate the zeta function for complex arguments.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 114px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 57px; transform-origin: 407px 57px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/45988\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problem 45988\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 315.85px 7.8px; transform-origin: 315.85px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e involved computing the Riemann zeta function for real arguments greater than 1. Code that works for that problem can reveal the connection between \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eπ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 222.1px 7.8px; transform-origin: 222.1px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and the values of the zeta function evaluated at positive even integers; this connection is explored in\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/45939-estimate-pi-from-certain-values-of-the-zeta-function\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCody Problem 45939\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 217.433px 7.8px; transform-origin: 217.433px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. However, to test the Riemann hypothesis--that all non-trivial zeros of the zeta function have a real part of 1/2, one needs to compute the zeta function for complex arguments.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 213.017px 7.8px; transform-origin: 213.017px 7.8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to evaluate the zeta function for complex arguments.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function z = zeta2(s)\r\n z = f(s);\r\nend","test_suite":"%%\r\ns = 2;\r\nz_correct = pi^2/6;\r\nassert(abs(zeta2(s)-z_correct)/z_correct \u003c 1e-8)\r\n\r\n%%\r\ns = 1;\r\nassert(isinf(zeta2(s)))\r\n\r\n%%\r\ns = 1/2;\r\nz_correct = -1.460354508809587;\r\nassert(abs((zeta2(s)-z_correct)/z_correct) \u003c 1e-8)\r\n\r\n%%\r\ns = 0;\r\nz_correct = -0.5;\r\nassert(abs((zeta2(s)-z_correct)/z_correct) \u003c 1e-8)\r\n\r\n%%\r\ns = -1;\r\nz_correct = -1/12;\r\nassert(abs((zeta2(s)-z_correct)/z_correct) \u003c 1e-8)\r\n\r\n%%\r\ns = -2;\r\nz_correct = 0;\r\nassert(abs(zeta2(s)) \u003c 1e-12)\r\n\r\n%%\r\ns = 3+2*i;\r\nz_correct = 0.973041960418942 - 0.147695593000454i;\r\nassert(abs(real(zeta2(s))-real(z_correct))/real(z_correct) \u003c 1e-8) \r\nassert(abs((imag(zeta2(s))-imag(z_correct))/imag(z_correct)) \u003c 1e-8)\r\n\r\n%%\r\ns = -1+2*i;\r\nz_correct = 0.168915669770846 - 0.070515988908259i;\r\nassert(abs((real(zeta2(s))-real(z_correct))/real(z_correct)) \u003c 1e-8) \r\nassert(abs((imag(zeta2(s))-imag(z_correct))/imag(z_correct)) \u003c 1e-8)\r\n\r\n%%\r\ns = 0.75-3*i;\r\nz_correct = 0.580900396083837 + 0.095281202690117i;\r\nassert(abs(real(zeta2(s))-real(z_correct))/real(z_correct) \u003c 1e-8) \r\nassert(abs((imag(zeta2(s))-imag(z_correct))/imag(z_correct)) \u003c 1e-8)\r\n\r\n%%\r\ns = 5+2*i;\r\nz_correct = 1.001916538615071 - 0.034217062736354i;\r\nassert(abs((real(zeta2(s))-real(z_correct))/real(z_correct)) \u003c 1e-8) \r\nassert(abs((imag(zeta2(s))-imag(z_correct))/imag(z_correct)) \u003c 1e-8)\r\n\r\n%%\r\ns = 0.5+14.13472514173469379*i;\r\nassert(abs(real(zeta2(s))) \u003c 1e-12) \r\nassert(abs(imag(zeta2(s))) \u003c 1e-12)\r\n\r\n%%\r\ns = 0.5+21*i;\r\nz_correct = -0.005162064638102 - 0.024546964575122i;\r\nassert(abs((real(zeta2(s))-real(z_correct))/real(z_correct)) \u003c 1e-8) \r\nassert(abs((imag(zeta2(s))-imag(z_correct))/imag(z_correct)) \u003c 1e-8)\r\n\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":3,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":12,"test_suite_updated_at":"2020-06-29T02:07:46.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2020-06-28T04:41:50.000Z","updated_at":"2026-01-09T13:36:37.000Z","published_at":"2020-06-28T05:14:18.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/45988\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem 45988\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e involved computing the Riemann zeta function for real arguments greater than 1. Code that works for that problem can reveal the connection between \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"pi\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and the values of the zeta function evaluated at positive even integers; this connection is explored in\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/45939-estimate-pi-from-certain-values-of-the-zeta-function\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody Problem 45939\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. However, to test the Riemann hypothesis--that all non-trivial zeros of the zeta function have a real part of 1/2, one needs to compute the zeta function for complex arguments.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to evaluate the zeta function for complex arguments.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":59521,"title":"Integrate a power tower","description":"Write a function to compute this integral\r\n\r\nwhere . That is, the integrand is (x to the x) to the (x to the x) to the (x to the x)...","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 104px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 52px; transform-origin: 407px 52px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 122.783px 8px; transform-origin: 122.783px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to compute this integral\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 44px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22px; text-align: left; transform-origin: 384px 22px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg 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AnGgx7jdhPsh4iVErvgcu4lMytGj7v56uakInAJxxMhe0XTq5tTcDXV+3D7Fyp+vuWVtajvpOSHQFAKnQBzxwiL39PSVRMkR+jEHMJzAbjWJ95g01cCqjBBYA4FTIA62KlyL6PeA+GVCd9p36R2afRi7y/k99qNfCeBkpNXGGr1SeTaPwCkQB43A4IckLptcNbnLxK+8K2kkzrNw6zuH27jR+9Emd1TmUVKO0giBXSBwKsSxi8ZQJYXAXhAQceylpVRPIdAQAiKOhhpDVRECe0FAxLGXllI9hUBDCIg4GmoMVUUI7AUBEcdeWkr1FAINISDiaKgxVBUhsBcE/g+tfmSG+LdlUAAAAABJRU5ErkJggg==\" width=\"135\" height=\"44\" alt=\"I = integral((x^x)^(x^x)^(x^x)...,{x,a,0})\" style=\"width: 135px; height: 44px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 8px; transform-origin: 21.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"63\" height=\"18\" style=\"width: 63px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 226.3px 8px; transform-origin: 226.3px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. That is, the integrand is (x to the x) to the (x to the x) to the (x to the x)...\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function I = intPowerTower(a)\r\n I = integral(x^x^x^x^x^x^x^x,0,a);\r\nend","test_suite":"%%\r\na = 0;\r\nI = intPowerTower(a);\r\nassert(abs(I)\u003c1e-6)\r\n\r\n%%\r\na = 1/100;\r\nI = intPowerTower(a);\r\nI_correct = 0.00975627404012066;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 1/20;\r\nI = intPowerTower(a);\r\nI_correct = 0.04621245261821598;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 1/10;\r\nI = intPowerTower(a);\r\nI_correct = 0.0886781687569094;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 1/5;\r\nI = intPowerTower(a);\r\nI_correct = 0.1685639964895788;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 1/4;\r\nI = intPowerTower(a);\r\nI_correct = 0.2071658901263798;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 3/8;\r\nI = intPowerTower(a);\r\nI_correct = 0.30215124860335973;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 1/2;\r\nI = intPowerTower(a);\r\nI_correct = 0.3972053202401857;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 2/3;\r\nI = intPowerTower(a);\r\nI_correct = 0.5277402852630483;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 3/4;\r\nI = intPowerTower(a);\r\nI_correct = 0.5959989560650945;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 5/6;\r\nI = intPowerTower(a);\r\nI_correct = 0.6671963910854818;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = 1;\r\nI = intPowerTower(a);\r\nI_correct = 0.822467033424113;\r\nassert(abs(I-I_correct)\u003c1e-8)\r\n\r\n%%\r\na = (rand+3)/4;\r\nI = intPowerTower(a);\r\nI_correct = polyval([0.3875275 -0.9886411 1.132527 0.1505356 0.1405179],a);\r\nassert(abs(I-I_correct)\u003c5e-6)\r\n\r\n%%\r\nfiletext = fileread('intPowerTower.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert') || contains(filetext,'regexp') || contains(filetext,'find') || contains(filetext,'switch'); \r\nassert(~illegal)\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":3,"created_by":46909,"edited_by":46909,"edited_at":"2024-01-03T15:06:22.000Z","deleted_by":null,"deleted_at":null,"solvers_count":7,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-12-31T18:50:11.000Z","updated_at":"2026-01-28T06:58:04.000Z","published_at":"2023-12-31T18:50:21.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to compute this integral\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"I = integral((x^x)^(x^x)^(x^x)...,{x,a,0})\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eI = \\\\int_0^a {(x^x)^{(x^x)^{(x^x)\\\\ldots}} dx\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e0 \\\\le a \\\\le 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. That is, the integrand is (x to the x) to the (x to the x) to the (x to the x)...\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51783,"title":"Solve an ODE: equation B","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 213.25px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 106.625px; transform-origin: 407px 106.625px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 211.358px 7.91667px; transform-origin: 211.358px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve the following ordinary differential equation: \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 39px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 19.5px; text-align: left; transform-origin: 384px 19.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg src=\"data:image/png;base64,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alt=\"y\u0026quot; + (1/x) y' - (a^2 + p^2/x^2) y = 0\" style=\"width: 181.5px; height: 39px;\" width=\"181.5\" height=\"39\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.25px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.125px; text-align: left; transform-origin: 384px 21.125px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14.3917px 7.91667px; transform-origin: 14.3917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewith \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(x1) = y1\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 35.0083px 7.91667px; transform-origin: 35.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and either \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(x0) = y0\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.1083px 7.91667px; transform-origin: 10.1083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e or \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJQAAAAoCAYAAAAYNNPaAAADoUlEQVR4nO2bbZWzMBCFrwccYKAGUFAFdVAHdYCFaqgEPGBhNdRC3x/hHrJsSPMxSWjfec7pnyVspnND5iMUUBRFURRFURTlq+kAnCLGnwCMhWxx0S/zdRXnPDKxelXnuXz6gLEnABPqizsAmBvMe0Ri9KrOGcALwE/A2NMyrpWoF5hF9T8To1cT7jAGXt+M62C+xLm4RX7uqBtuj0aoXs14ImzXucOEutZ0MDYPrQ1pRKheTTghbLX3y7ijiDjiwFt+QUL1agaFCdmdjpS7cIG3Dr+1CdWrGTPei9LBiHcrb04UM4BHayMqE6JXNh1MKDrDiO4qJXnN7l30CNt1LogLd5013/aeDma7llic42LXp1FaryyGZeIJxrkvuCsgXttulyF9DFYV7+hhdowJJnHczney/v6CWag5xC50G/ot9xPbB6qhlxg08om/MfbiuRbyf5+R99ywOmXA2nLg7jQh3zkD0hNU276cT063upReYrDhtff0U8hYXgn3sRqh4BPkKxPOkRI+pXaoHLFL6SUKQ8rdce2ONFFTFpRty3PHHgle+OzEvIReojDf2cZehpyUMJO6oB5Yn0DfvB2M3fxMCA8lqbYdBWm9cnzphLF3m6zekH5ckSra1bJlLzSw6/2wxvC+EEd8+oKS1CvXl07Y8LOT1ZzdCUgXjU+fr6LjmG2l9sT7EpnftWUOlVtYSOqV40svP/idW4zIO0ydEwy6LHb4cgQ2TF07GCsgX0uAVV7KgjpClUck9Mr1pReu1Blr7yenGmEuFApfcRmse13nblwQrsVKwX2OZZXUsg8lUdJL6JXrSy92XJ6R30CMicPdZk47j+IW/sDaCX7nBF+o5ZjDnmsFIqFXri+92D0giYSVcX7vi45Y+0wzfj8Jti0j1uoDCHOCL9SmhOIjIqFXri/fwv6P1HvHvoNYO6dwjfnZuZ7jBOYMubvvUcjVq8qCkmyKcVt2VR7MR/bE7XeuMxymOOGKxscSwuTqlePLoH9eons8Q/YVlpxEkueC34CEXsWS8gHlXsBirJc88WYo3MJS1/Xuzw3fkTsBsnql+NILy/WSv9eS/tWJrxnncjTL6kP+fCgSab1iffkHvmA2YW0k1njn+wa5w14eF9jVDfOB7RPFlsShf+DoobReMb50Yh++lt6Ztlwgt6h6rAeZI8z3cr3p+cDnLiagjl4hvtyFr9O2Kp1rVlhd5flK0FovRVEURVEURVEURVGUEP4BkX7JQnEwa3oAAAAASUVORK5CYII=\" alt=\"y'(x0) = y'0\" style=\"width: 74px; height: 20px;\" width=\"74\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 38.1167px 7.91667px; transform-origin: 38.1167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Along with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.7px 7.91667px; transform-origin: 7.7px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003ey1\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 25.275px 7.91667px; transform-origin: 25.275px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, one of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.7px 7.91667px; transform-origin: 7.7px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003ey0\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.55px 7.91667px; transform-origin: 11.55px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eyp0\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 109.708px 7.91667px; transform-origin: 109.708px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e will be assigned numerical values, and the other will be \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.55px 7.91667px; transform-origin: 11.55px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eNaN\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 128.742px 7.91667px; transform-origin: 128.742px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The function should return the values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ey\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 80.9px 7.91667px; transform-origin: 80.9px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at the specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 194.483px 7.91667px; transform-origin: 194.483px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eOne of the applications for this equation is in groundwater. For \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"a = 1\" style=\"width: 36.5px; height: 18px;\" width=\"36.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p = 0\" style=\"width: 38px; height: 18px;\" width=\"38\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 136.567px 7.91667px; transform-origin: 136.567px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e the equation arises in the flow of water to a well pumping in a leaky confined aquifer; the independent variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 141.975px 7.91667px; transform-origin: 141.975px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the normalized distance from the well, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ey\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 8.94167px 7.91667px; transform-origin: 8.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is related to the piezometric head, a combination of the elevation and pressure of the water. Specifying the derivative at a point amounts to specifying the flow to the well.\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = solveODEB(x,coeff,Xbc,bc)\r\n% y = values of the solution\r\n% x = values of the independent variable where the solution is requested\r\n% coeff = [a p], the two parameters in the ODE\r\n% Xbc = values of x where the boundary conditions are specifified\r\n% bc = [y0 yp0 y1] = boundary values (see description)\r\n\r\n y = f(x,coeff,Xbc,bc);\r\nend","test_suite":"%%\r\ncoeff = [1 0];\r\nXbc = [1 4];\r\nbc = [1 NaN 0];\r\nx = linspace(1,4,7);\r\ny = solveODEB(x,coeff,Xbc,bc);\r\ny_correct = [1 0.505461057363874 0.265959532078956 0.140787844664873 0.071276733472728 0.029333752731051 0];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%%\r\ncoeff = [3 1];\r\nXbc = [0.5 4.5];\r\nbc = [0 NaN 2];\r\nx = [cos(pi/4) cos(pi/6) sqrt(2) (1+sqrt(5))/2 sqrt(3) sqrt(5) exp(1) pi 4+psi(1)];\r\ny = solveODEB(x,coeff,Xbc,bc);\r\ny_correct = [0.000035290165611 0.000065476121971 0.000315436935125 0.000552779648598 0.000757412623201 0.003086031722519 0.012031119481478 0.040129255417251 0.089700339919646];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%%\r\ncoeff = [1 1/2];\r\nXbc = [1 5];\r\nbc = [4 NaN -1];\r\nx = 1.2:0.75:4.95;\r\ny = solveODEB(x,coeff,Xbc,bc);\r\ny_correct = [2.974028994378906 1.041176175714286 0.308462205553512 -0.076175488441917 -0.426089583908447 -0.952692417292867];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%%\r\ncoeff = [2 4];\r\nXbc = [1 Inf];\r\nbc = [3 NaN 0];\r\nx = 1:2:9;\r\ny = solveODEB(x,coeff,Xbc,bc);\r\ny_correct = [3 0.005688558853080 0.000051725255081 0.000000653418086 0.000000009396692];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n%-------------------------------\r\n%%\r\ncoeff = [1 0];\r\nXbc = [1 4];\r\nbc = [NaN 1 0];\r\nx = linspace(1,4,7);\r\ny = solveODEB(x,coeff,Xbc,bc);\r\ny_correct = [-0.696760997897786 -0.352185550727323 -0.185310228971762 -0.098095479140575 -0.049662847941353 -0.020438614824974 0];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n \r\n%%\r\ncoeff = [3 1];\r\nXbc = [0.5 4.5];\r\nbc = [NaN 0 2];\r\nx = [cos(pi/4) cos(pi/6) sqrt(2) (1+sqrt(5))/2 sqrt(3) sqrt(5) exp(1) pi 4+psi(1)];\r\ny = solveODEB(x,coeff,Xbc,bc);\r\ny_correct = [0.000053997354167 0.000075728700374 0.000316920386259 0.000553525214653 0.000757922298088 0.003086129240342 0.012031140116004 0.040129260776539 0.089700342119009];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%%\r\ncoeff = [1 1/2];\r\nXbc = [1 5];\r\nbc = [NaN 4 -1];\r\nx = 1.2:0.75:4.95;\r\ny = solveODEB(x,coeff,Xbc,bc);\r\ny_correct = [-2.047695617799804 -0.816404083826666 -0.431398160565887 -0.374418157507686 -0.532801281305956 -0.958228725044217];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%%\r\ncoeff = [1 0];\r\nXbc = [1 Inf];\r\nbc = [NaN 3 0];\r\nx = 1:2:9;\r\ny = solveODEB(x,coeff,Xbc,bc);\r\ny_correct = [-2.098451806781317 -0.173147136186896 -0.018397012773047 -0.002117248575316 -0.000253600440751];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":2,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":8,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-05-16T01:10:22.000Z","updated_at":"2025-05-09T06:39:10.000Z","published_at":"2021-05-16T01:19:31.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve the following ordinary differential equation: \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y\u0026quot; + (1/x) y' - (a^2 + p^2/x^2) y = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\frac{d^2y}{dx^2} +\\\\frac{1}{x} \\\\frac{dy}{dx}-\\\\left(a^2+\\\\frac{p^2}{x^2}\\\\right)y = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewith \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(x1) = y1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(x_1) = y_1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and either \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(x0) = y0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(x_0) = y_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e or \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y'(x0) = y'0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\\\\prime(x_0) = y\\\\prime_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Along with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey1\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, one of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eyp0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e will be assigned numerical values, and the other will be \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eNaN\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. The function should return the values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at the specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOne of the applications for this equation is in groundwater. For \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a = 1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea = 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e the equation arises in the flow of water to a well pumping in a leaky confined aquifer; the independent variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the normalized distance from the well, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is related to the piezometric head, a combination of the elevation and pressure of the water. Specifying the derivative at a point amounts to specifying the flow to the well.\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":46025,"title":"Evaluate the gamma function","description":"The gamma function is a generalization of the factorial, and it appears in many applications such as evaluating certain integrals, working with probability distributions, and evaluating fractional derivatives. MATLAB includes the function gamma, but it accepts only real arguments.\r\nWrite a function that evaluates the gamma function for complex arguments.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 93.3333px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407.5px 46.6667px; transform-origin: 407.5px 46.6667px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63.3333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 31.6667px; text-align: left; transform-origin: 384.5px 31.6667px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12.0583px 7.66667px; transform-origin: 12.0583px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.66667px; transform-origin: 1.94167px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://mathworld.wolfram.com/GammaFunction.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003egamma function\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 302.642px 7.66667px; transform-origin: 302.642px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a generalization of the factorial, and it appears in many applications such as evaluating certain integrals, working with probability distributions, and evaluating fractional derivatives. MATLAB includes the function\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.66667px; transform-origin: 1.94167px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 19.25px 7.66667px; transform-origin: 19.25px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 19.25px 8px; transform-origin: 19.25px 8px; \"\u003egamma\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.66667px; transform-origin: 3.88333px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, but it accepts only real arguments.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 232.467px 7.66667px; transform-origin: 232.467px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that evaluates the gamma function for complex arguments.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = gamma2(z)\r\n y = f(z);\r\nend","test_suite":"%%\r\nz = 3+2i;\r\ny = gamma2(z);\r\ny_correct = -0.4226372863112003 + 0.871814255696503i;\r\nassert(abs((real(y)-real(y_correct))/real(y_correct)) \u003c 1e-6)\r\nassert(abs((imag(y)-imag(y_correct))/imag(y_correct)) \u003c 1e-6)\r\n\r\n%%\r\nz = 1+i;\r\ny = gamma2(z);\r\ny_correct = 0.4980156681183556 -0.1549498283018104i;\r\nassert(abs((real(y)-real(y_correct))/real(y_correct)) \u003c 1e-6)\r\nassert(abs((imag(y)-imag(y_correct))/imag(y_correct)) \u003c 1e-6)\r\n\r\n%%\r\nz = (1+i)/2;\r\ny = gamma2(z);\r\ny_correct = 0.818163995 - 0.7633138287i;\r\nassert(abs((real(y)-real(y_correct))/real(y_correct)) \u003c 1e-6)\r\nassert(abs((imag(y)-imag(y_correct))/imag(y_correct)) \u003c 1e-6)\r\n\r\n%%\r\nz = i;\r\ny = gamma2(z);\r\ny_correct = -0.154949828301810 - 0.4980156681183566i;\r\nassert(abs((real(y)-real(y_correct))/real(y_correct)) \u003c 1e-6)\r\nassert(abs((imag(y)-imag(y_correct))/imag(y_correct)) \u003c 1e-6)\r\n\r\n%%\r\nz = 5i;\r\ny = gamma2(z);\r\ny_correct = -0.00027170388350615125 + 0.0003399328988721375i;\r\nassert(abs((real(y)-real(y_correct))/real(y_correct)) \u003c 1e-6)\r\nassert(abs((imag(y)-imag(y_correct))/imag(y_correct)) \u003c 1e-6)\r\n\r\n%%\r\nz = 1/2 + 14.1i;\r\ny = gamma2(z);\r\ny_correct = -2.0555298837259187e-10 - 5.667644214210669e-10i;\r\nassert(abs((real(y)-real(y_correct))/real(y_correct)) \u003c 1e-6)\r\nassert(abs((imag(y)-imag(y_correct))/imag(y_correct)) \u003c 1e-6)\r\n\r\n%%\r\nz = -1+i;\r\ny = gamma2(z);\r\ny_correct = -0.1715329199082727 + 0.3264827482100833i;\r\nassert(abs((real(y)-real(y_correct))/real(y_correct)) \u003c 1e-6)\r\nassert(abs((imag(y)-imag(y_correct))/imag(y_correct)) \u003c 1e-6)\r\n\r\n%%\r\nz = -2-3i;\r\ny = gamma2(z);\r\ny_correct = -0.0001631724182726072 - 0.001128495917017955i;\r\nassert(abs((real(y)-real(y_correct))/real(y_correct)) \u003c 1e-6)\r\nassert(abs((imag(y)-imag(y_correct))/imag(y_correct)) \u003c 1e-6)\r\n\r\n%%\r\nz = 10*(rand+0.02);\r\ny_correct = gamma(z);\r\nassert(abs(gamma2(z)-y_correct)/y_correct \u003c 1e-6)\r\n\r\n%%\r\nfiletext = fileread('gamma2.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'regexp') || contains(filetext, 'system'); \r\nassert(~illegal)\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":9,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":20,"test_suite_updated_at":"2022-01-30T20:31:21.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2020-07-04T16:14:47.000Z","updated_at":"2026-01-09T11:39:54.000Z","published_at":"2020-07-05T04:43:57.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://mathworld.wolfram.com/GammaFunction.html\\\"\u003e\u003cw:r\u003e\u003cw:t\u003egamma function\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e is a generalization of the factorial, and it appears in many applications such as evaluating certain integrals, working with probability distributions, and evaluating fractional derivatives. MATLAB includes the function\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003egamma\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, but it accepts only real arguments.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that evaluates the gamma function for complex arguments.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":59541,"title":"Compute the polylogarithm for real arguments","description":"The polylogarithm appears in quantum statistics and quantum electrodynamics, and for special cases of and , it connects to the logarithm, ratios of polynomials, the zeta function, the Dirichlet eta function, and other functions. \r\nWrite a function to compute the polylogarithm for real arguments. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 74px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 37px; transform-origin: 407px 37px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 43px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.5px; text-align: left; transform-origin: 384px 21.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 57.575px 8px; transform-origin: 57.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe polylogarithm \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"39\" height=\"20\" alt=\"Li_n(z)\" style=\"width: 39px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 267.225px 8px; transform-origin: 267.225px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e appears in quantum statistics and quantum electrodynamics, and for special cases of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ez\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 9.325px 8px; transform-origin: 9.325px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, it connects to the logarithm, ratios of polynomials, the \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/45988\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003ezeta\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/45997\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003efunction\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 146.242px 8px; transform-origin: 146.242px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, the Dirichlet eta function, and other functions. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 11px; text-align: left; transform-origin: 384px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 142.233px 8px; transform-origin: 142.233px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to compute the polylogarithm \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"39\" height=\"20\" alt=\"Li_n(z)\" style=\"width: 39px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 62.6167px 8px; transform-origin: 62.6167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e for real arguments. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = polylogarithm(n,z)\r\n y = sum(z^k/k^n);\r\nend","test_suite":"%%\r\nn = 2:2:16;\r\nz = 1;\r\ny = arrayfun(@(m) polylogarithm(m,z),n);\r\ny_correct = pi.^n.*[1/6 1/90 1/945 1/9450 1/93555 691/638512875 2/18243225 3617/325641566250];\r\nassert(all(abs(y-y_correct)\u003c1e-12))\r\n\r\n%%\r\nn = 3;\r\nz = 1;\r\ny = polylogarithm(n,z);\r\ny_correct = 1.202056903159594;\r\nassert(abs(y-y_correct)\u003c1e-12)\r\n\r\n%%\r\nn = 0;\r\nz = 1./(10:-1:2);\r\ny = arrayfun(@(s) polylogarithm(n,s),z);\r\ny_correct = 1./(9:-1:1);\r\nassert(all(abs(y-y_correct)\u003c1e-12))\r\n\r\n%%\r\nn = 1;\r\nz = [-2 -3/2 -1 -1/2 0 1/10 1/7 1/5 1/3 1/2];\r\ny = arrayfun(@(s) polylogarithm(n,s),z);\r\ny_correct = [-1.098612288668110 -0.916290731874155 -0.693147180559945 -0.405465108108164 0 0.105360515657826 0.154150679827258 0.223143551314210 0.405465108108164 0.693147180559945];\r\nassert(all(abs(y-y_correct)\u003c1e-12))\r\n\r\n%%\r\nn = -1;\r\nz = [-2 -3/2 -1 -1/2 1/7 1/5 1/2];\r\ny = arrayfun(@(s) polylogarithm(n,s),z);\r\ny_correct = [-2/9 -6/25 -1/4 -2/9 7/36 5/16 2];\r\nassert(all(abs(y-y_correct)\u003c1e-12))\r\n\r\n%%\r\nn = -2;\r\nz = [-2 -3/2 -1 -1/2 0 1/17 1/13 1/11 1/5];\r\ny = arrayfun(@(s) polylogarithm(n,s),z);\r\ny_correct = [2/27 6/125 0 -2/27 0 153/2048 91/864 33/250 15/32];\r\nassert(all(abs(y-y_correct)\u003c1e-12))\r\n\r\n%%\r\nn = 1:3;\r\nz = 1/2;\r\ny = arrayfun(@(m) polylogarithm(m,z),n);\r\nzeta3 = 1.202056903159594;\r\ny_correct = [log(2) (pi^2-6*log(2)^2)/12 (4*log(2)^3-2*pi^2*log(2)+21*zeta3)/24];\r\nassert(all(abs(y-y_correct)\u003c1e-12))\r\n\r\n%%\r\nn = 2;\r\nz = -1;\r\ny = polylogarithm(n,z);\r\ny_correct = -pi^2/12;\r\nassert(abs(y-y_correct)\u003c1e-12)\r\n\r\n%%\r\nn = -1/pi;\r\nz = exp(-1);\r\ny = polylogarithm(n,z);\r\ny_correct = 0.6541056465726233;\r\nassert(abs(y-y_correct)\u003c1e-12)\r\n\r\n%%\r\nn = 4;\r\nz = 1/2;\r\ny = polylogarithm(n,z);\r\ny_correct = 0.5174790616738994;\r\nassert(abs(y-y_correct)\u003c1e-12)\r\n\r\n%%\r\nn = 1/2;\r\nz = 1/2;\r\ny = polylogarithm(n,z);\r\ny_correct = 0.8061267230428522;\r\nassert(abs(y-y_correct)\u003c1e-12)\r\n\r\n%%\r\nn = -3/2;\r\nz = -1./2.^(1:4);\r\ny = arrayfun(@(s) polylogarithm(n,s),z);\r\ny_correct = [-0.1516441361365191 -0.1313580923579523 -0.0892942267818043 -0.05260782861980105];\r\nassert(all(abs(y-y_correct)\u003c1e-12))\r\n\r\n%%\r\nn = 5*randn(1,randi(10));\r\nz = 0;\r\ny = arrayfun(@(m) polylogarithm(m,z),n);\r\ny_correct = zeros(size(n));\r\nassert(all(abs(y-y_correct)\u003c1e-12))\r\n\r\n%%\r\nfiletext = fileread('polylogarithm.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert') || contains(filetext,'regexp'); \r\nassert(~illegal)","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2024-01-07T03:51:00.000Z","deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2024-01-06T17:30:34.000Z","updated_at":"2025-02-27T16:14:20.000Z","published_at":"2024-01-06T17:31:22.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe polylogarithm \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Li_n(z)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e{\\\\rm Li}_n(z)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e appears in quantum statistics and quantum electrodynamics, and for special cases of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ez\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, it connects to the logarithm, ratios of polynomials, the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/45988\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ezeta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink 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