# dawson

Dawson integral

## Description

example

dawson(X) represents the Dawson integral.

## 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

### 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

collapse all

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

collapse all

### Dawson Integral

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

$\text{dawson}\left(x\right)=D\left(x\right)={e}^{-{x}^{2}}\underset{0}{\overset{x}{\int }}{e}^{{t}^{2}}dt$

Symbolic Math Toolbox™ uses this definition to implement dawson.

The alternative definition of the Dawson integral is

$D\left(x\right)={e}^{{x}^{2}}\underset{0}{\overset{x}{\int }}{e}^{-{t}^{2}}dt$

## Tips

• dawson(0) returns 0.

• dawson(Inf) returns 0.

• dawson(-Inf) returns 0.

## Version History

Introduced in R2014a