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.

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  (25)

Numerical Evaluation  (4)

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:

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  (5)

DawsonF is defined for all real and complex values:

Approximate function range for DawsonF:

DawsonF is an odd function:

DawsonF has the mirror property TemplateBox[{{z, }}, DawsonF]=TemplateBox[{z}, DawsonF]:

DawsonF threads elementwise over lists:

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 ^(th) 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  (2)

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

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

Properties & Relations  (1)

Use FunctionExpand to expand DawsonF in terms of error function:

Introduced in 2008
 (7.0)