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# dawson

Dawson integral

• dawson(X)

## Description

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
ezplot(dawson(x), [-10, 10])
grid on
```

### Handle Expressions Containing the 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

expand all

### X — Inputsymbolic 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.

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### 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.