DawsonF[z]
gives the Dawson integral
.
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
open all close allBasic Examples (5)
DawsonF[2.5]Plot over a subset of the reals:
Plot[DawsonF[x], {x, -10, 10}]Plot over a subset of the complexes:
ComplexPlot3D[DawsonF[z], {z, -2 - 2I, 2 + 2I}, PlotLegends -> Automatic]Series expansion at the origin:
Series[DawsonF[x], {x, 0, 10}]Asymptotic expansion at Infinity:
Series[DawsonF[x], {x, ∞, 5}]//Normal//FullSimplifyScope (33)
Numerical Evaluation (6)
N[DawsonF[1 / E]]DawsonF[-.46]N[DawsonF[4 / 3], 50]The precision of the output tracks the precision of the input:
DawsonF[1.11111111111111111111111]N[DawsonF[I Pi + 5]]Evaluate efficiently at high precision:
DawsonF[.9`100]//TimingDawsonF[58.`10000];//TimingCompute worst-case guaranteed intervals using Interval and CenteredInterval objects:
DawsonF[Interval[{0.5, 0.6}]]DawsonF[CenteredInterval[1 / 2, 1 / 100]]Or compute average-case statistical intervals using Around:
DawsonF[ Around[2.1, 0.01]]Compute the elementwise values of an array:
DawsonF[{{1.2, 0}, {0, -1.2}}]Or compute the matrix DawsonF function using MatrixFunction:
MatrixFunction[DawsonF, {{1.2, 0}, {0, -1.2}}]Specific Values (3)
Simple exact values are generated automatically:
DawsonF[0]{Limit[DawsonF[x], x -> Infinity], Limit[DawsonF[x], x -> -Infinity]}Find positive maximum of DawsonF[x]:
xmax = x /. FindRoot[D[DawsonF[x ], x] == 0, {x, 1}]Plot[DawsonF[x ], {x, -5, 5}, Epilog -> Style[Point[{xmax, DawsonF[xmax ]}], PointSize[Large], Red]]Visualization (2)
Plot the DawsonF function:
Plot[DawsonF[x], {x, -10, 10}]Plot the real part of DawsonF:
ComplexContourPlot[Re[DawsonF[z]], {z, -4 - 4 I, 4 + 4 I}, Contours -> 20]Plot the imaginary part of DawsonF:
ComplexContourPlot[Im[DawsonF[z]], {z, -4 - 4 I, 4 + 4 I}, Contours -> 20]Function Properties (11)
DawsonF is defined for all real and complex values:
FunctionDomain[DawsonF[x], x]FunctionDomain[DawsonF[z], z, Complexes]Approximate function range for DawsonF:
FunctionRange[DawsonF[x], x, y]//QuietDawsonF is an odd function:
DawsonF[-x]DawsonF has the mirror property
:
FullSimplify[DawsonF[Conjugate[z]] == Conjugate[DawsonF[z]]]DawsonF threads elementwise over lists:
DawsonF[{1.0, 2.0, 3.0, 4.0, 5.0}]DawsonF is an analytic function of x:
FunctionAnalytic[DawsonF[x], x]Has no singularities or discontinuities:
FunctionSingularities[DawsonF[x], x]//QuietFunctionDiscontinuities[DawsonF[x], x]//QuietDawsonF is neither nondecreasing nor nonincreasing:
FunctionMonotonicity[DawsonF[x], x]DawsonF is not injective:
FunctionInjective[DawsonF[x], x]Plot[{DawsonF[x], .2}, {x, -10, 10}]DawsonF is not surjective:
FunctionSurjective[DawsonF[x], x]Plot[{DawsonF[x], -1}, {x, -10, 10}]DawsonF is neither non-negative nor non-positive:
FunctionSign[DawsonF[x], x]DawsonF is neither convex nor concave:
FunctionConvexity[DawsonF[x], x]Differentiation (3)
First derivative with respect to z:
D[DawsonF[z], z]Higher derivatives with respect to z:
Table[D[DawsonF[z], {z, k}], {k, 1, 3}]//FullSimplifyPlot the higher derivatives with respect to z:
Plot[%, {z, -4, 4}, PlotLegends -> {"First Derivative", "Second Derivative", "Third Derivative"}]Formula for the ![]()
derivative with respect to
:
D[DawsonF[z], {z, k}]// FullSimplifyIntegration (3)
Compute the indefinite integral using Integrate:
Integrate[DawsonF[x], x]// FullSimplifyFullSimplify[D[%, x]]// FullSimplifyIntegrate[DawsonF[x], {x, 0, 15}]Integrate[DawsonF[x] x ^ a, x]// FullSimplifyIntegrate[DawsonF[x + a] x, x]// FullSimplifySeries Expansions (5)
Find the Taylor expansion using Series:
Series[DawsonF[x], {x, 0, 10}]Plots of the first three approximations around
:
terms = Normal@Table[Series[DawsonF[x], {x, 0, m}], {m, 1, 5, 2}];
Plot[{DawsonF[x], terms}, {x, -5, 5}, PlotRange -> {-10, 10}]General term in the series expansion using SeriesCoefficient:
SeriesCoefficient[DawsonF[x], {x, 1, n}]Find the series expansion at Infinity:
Series[DawsonF[x], {x, Infinity, 1}]Find series expansion for an arbitrary symbolic direction
:
Series[DawsonF[x], {x, DirectedInfinity[z], 1}, Assumptions -> x > 0]// FullSimplifyTaylor expansion at a generic point:
Series[DawsonF[x], {x, x0, 2}]// FullSimplifyGeneralizations & Extensions (2)
DawsonF can be applied to a power series:
DawsonF[ Log[1 + x] + O[x] ^ 10]Infinite arguments give symbolic results:
DawsonF[{-Infinity, Infinity, I Infinity, -I Infinity}]Applications (3)
Find the value and position of the maximum of the Dawson function:
FindMaximum[DawsonF[x], {x, 1}]Express a probability density function in terms of the Dawson function:
𝒟 = TransformedDistribution[u ^ 2 + 3v, {u, v} ProductDistribution[{ExponentialDistribution[3], 2}]];PDF[𝒟, x]DawsonF appears in the Fourier transform of truncated Gaussians:
ft = FourierTransform[Exp[-x^2]UnitStep[x], x, ω]ComplexPlot3D[Evaluate[ft], {ω, -10 - 10I, 10 + 0I}, ScalingFunctions -> "Log"]Properties & Relations (1)
Use FunctionExpand to expand DawsonF in terms of the imaginary error function:
FunctionExpand[DawsonF[x]]Related Guides
History
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_2026_dawsonf, author="Wolfram Research", title="{DawsonF}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/DawsonF.html}", note=[Accessed: 12-June-2026]}
BibLaTeX
@online{reference.wolfram_2026_dawsonf, organization={Wolfram Research}, title={DawsonF}, year={2008}, url={https://reference.wolfram.com/language/ref/DawsonF.html}, note=[Accessed: 12-June-2026]}