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logint

Logarithmic integral function

Description

example

A = logint(x) evaluates the logarithmic integral function (integral logarithm).

Examples

Integral Logarithm for Numeric and Symbolic Arguments

logint returns floating-point or exact symbolic results depending on the arguments you use.

Compute integral logarithms for these numbers. Because these numbers are not symbolic objects, logint returns floating-point results.

A = logint([-1, 0, 1/4, 1/2, 1, 2, 10])
A =
   0.0737 + 3.4227i   0.0000 + 0.0000i  -0.1187 + 0.0000i  -0.3787 + 0.0000i...
     -Inf + 0.0000i   1.0452 + 0.0000i   6.1656 + 0.0000i

Compute integral logarithms for the numbers converted to symbolic objects. For many symbolic (exact) numbers, logint returns unresolved symbolic calls.

symA = logint(sym([-1, 0, 1/4, 1/2, 1, 2, 10]))
symA =
[ logint(-1), 0, logint(1/4), logint(1/2), -Inf, logint(2), logint(10)]

Use vpa to approximate symbolic results with floating-point numbers:

A = vpa(symA)
A =
[ 0.07366791204642548599010096523015...
 + 3.4227333787773627895923750617977i,...
0,...
-0.11866205644712310530509570647204,...
-0.37867104306108797672720718463656,...
-Inf,...
1.0451637801174927848445888891946,...
6.1655995047872979375229817526695]

Plot Integral Logarithm

Plot the integral logarithm function on the interval from 0 to 10.

syms x
fplot(logint(x),[0 10])
grid on

Figure contains an axes object. The axes object contains an object of type functionline.

Handle Expressions Containing Integral Logarithm

Many functions, such as diff and limit, can handle expressions containing logint.

Find the first and second derivatives of the integral logarithm:

syms x
dA = diff(logint(x), x)
dA = diff(logint(x), x, x)
dA =
1/log(x)
 
dA =
-1/(x*log(x)^2)

Find the right and left limits of this expression involving logint:

A_r = limit(exp(1/x)/logint(x + 1), x, 0, 'right')
A_r =
Inf
A_l = limit(exp(1/x)/logint(x + 1), x, 0, 'left')
A_l =
0

Input Arguments

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Input, specified as a symbolic number, variable, expression, or function, or as a vector or matrix of symbolic numbers, variables, expressions, or functions.

More About

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Logarithmic Integral Function

The logarithmic integral function, also called the integral logarithm, is defined as follows:

logint(x)=li(x)=0x1ln(t)dt

Tips

  • logint(sym(0)) returns 1.

  • logint(sym(1)) returns -Inf.

  • logint(z) = ei(log(z)) for all complex z.

References

[1] Gautschi, W., and W. F. Cahill. “Exponential Integral and Related Functions.” Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. (M. Abramowitz and I. A. Stegun, eds.). New York: Dover, 1972.

See Also

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Introduced in R2014a