# DawsonF

DawsonF[z]

gives the Dawson integral .

# Details

• Mathematical function, suitable for both symbolic and numerical manipulation.
• The Dawson integral is defined by .
• For certain special arguments, DawsonF automatically evaluates to exact values.
• DawsonF can be numerically evaluated to arbitrary numerical precision.
• DawsonF automatically threads over lists.
• DawsonF can be used with Interval and CenteredInterval objects. »

# Examples

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## Basic Examples(5)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Asymptotic expansion at Infinity:

## Scope(32)

### Numerical Evaluation(5)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number inputs:

Evaluate efficiently at high precision:

DawsonF can be used with Interval and CenteredInterval objects:

### Specific Values(3)

Simple exact values are generated automatically:

Limiting values at infinity:

Find positive maximum of DawsonF[x]:

### Visualization(2)

Plot the DawsonF function:

Plot the real part of DawsonF:

Plot the imaginary part of DawsonF:

### Function Properties(11)

DawsonF is defined for all real and complex values:

Approximate function range for DawsonF:

DawsonF is an odd function:

DawsonF has the mirror property :

DawsonF threads elementwise over lists:

DawsonF is an analytic function of x:

Has no singularities or discontinuities:

DawsonF is neither nondecreasing nor nonincreasing:

DawsonF is not injective:

DawsonF is not surjective:

DawsonF is neither non-negative nor non-positive:

DawsonF is neither convex nor concave:

### Differentiation(3)

First derivative with respect to z:

Higher derivatives with respect to z:

Plot the higher derivatives with respect to z:

Formula for the derivative with respect to :

### Integration(3)

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

More integrals:

### Series Expansions(5)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

General term in the series expansion using SeriesCoefficient:

Find the series expansion at Infinity:

Find series expansion for an arbitrary symbolic direction :

Taylor expansion at a generic point:

## Generalizations & Extensions(2)

DawsonF can be applied to a power series:

Infinite arguments give symbolic results:

## Applications(3)

Find the value and position of the maximum of the Dawson function:

Express a probability density function in terms of the Dawson function:

DawsonF appears in the Fourier transform of truncated Gaussians:

Visualize the transform:

## Properties & Relations(1)

Use FunctionExpand to expand DawsonF in terms of the imaginary error function:

Wolfram Research (2008), DawsonF, Wolfram Language function, https://reference.wolfram.com/language/ref/DawsonF.html.

#### Text

Wolfram Research (2008), DawsonF, Wolfram Language function, https://reference.wolfram.com/language/ref/DawsonF.html.

#### CMS

Wolfram Language. 2008. "DawsonF." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/DawsonF.html.

#### APA

Wolfram Language. (2008). DawsonF. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DawsonF.html

#### BibTeX

@misc{reference.wolfram_2024_dawsonf, author="Wolfram Research", title="{DawsonF}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/DawsonF.html}", note=[Accessed: 20-May-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_dawsonf, organization={Wolfram Research}, title={DawsonF}, year={2008}, url={https://reference.wolfram.com/language/ref/DawsonF.html}, note=[Accessed: 20-May-2024 ]}