dawson

Dawson integral

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Syntax

dawson(X)

Description

dawson(X) represents the Dawson integral.

example

Examples

Dawson Integral for Numeric and Symbolic Arguments

Depending on its arguments, dawson returns floating-point or exact symbolic results.

Compute the Dawson integrals for these numbers. Because these numbers are not symbolic objects, dawson returns floating-point results.

A = dawson([-Inf, -3/2, -1, 0, 2, Inf])

A =
         0   -0.4282   -0.5381         0    0.3013         0

Compute the Dawson integrals for the numbers converted to symbolic objects. For many symbolic (exact) numbers, dawson returns unresolved symbolic calls.

symA = dawson(sym([-Inf, -3/2, -1, 0, 2, Inf]))

symA =
[ 0, -dawson(3/2), -dawson(1), 0, dawson(2), 0]

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

vpa(symA)

ans =
[ 0,...
-0.42824907108539862547719010515175,...
-0.53807950691276841913638742040756,...
0,...
0.30134038892379196603466443928642,...
0]

Plot the Dawson Integral

Plot the Dawson integral on the interval from -10 to 10.

syms x
fplot(dawson(x),[-10 10])
grid on

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

Handle Expressions Containing Dawson Integral

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

Find the first and second derivatives of the Dawson integral:

syms x
diff(dawson(x), x)
diff(dawson(x), x, x)

ans =
1 - 2*x*dawson(x)
 
ans =
2*x*(2*x*dawson(x) - 1) - 2*dawson(x)

Find the limit of this expression involving dawson:

limit(x*dawson(x), Inf)

ans =
1/2

Input Arguments

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`X` — Input
symbolic number | symbolic variable | symbolic expression | symbolic function | symbolic vector | symbolic matrix

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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Dawson Integral

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

$dawson (x) = D (x) = e^{- x^{2}} \int_{0}^{x} e^{t^{2}} d t$

Symbolic Math Toolbox™ uses this definition to implement dawson.

The alternative definition of the Dawson integral is

$D (x) = e^{x^{2}} \int_{0}^{x} e^{- t^{2}} d t$

Tips

dawson(0) returns 0.
dawson(Inf) returns 0.
dawson(-Inf) returns 0.

Version History

Introduced in R2014a