# Dawson function

In mathematics, the Dawson function or Dawson integral (named for H. G. Dawson[1]) is either

${\displaystyle F(x)=D_{+}(x)=e^{-x^{2}}\int _{0}^{x}e^{t^{2}}\,dt}$,

also denoted as F(x) or D(x), or alternatively

${\displaystyle D_{-}(x)=e^{x^{2}}\int _{0}^{x}e^{-t^{2}}\,dt\!}$.

The Dawson function is the one-sided Fourier-Laplace sine transform of the Gaussian function,

${\displaystyle D_{+}(x)={\frac {1}{2}}\int _{0}^{\infty }e^{-t^{2}/4}\,\sin {(xt)}\,dt.}$

It is closely related to the error function erf, as

${\displaystyle D_{+}(x)={{\sqrt {\pi }} \over 2}e^{-x^{2}}{\mathrm {erfi} }(x)=-{i{\sqrt {\pi }} \over 2}e^{-x^{2}}{\mathrm {erf} }(ix)}$

where erfi is the imaginary error function, erfi(x) = −i erf(ix). Similarly,

${\displaystyle D_{-}(x)={\frac {\sqrt {\pi }}{2}}e^{x^{2}}{\mathrm {erf} }(x)}$

in terms of the real error function, erf.

In terms of either erfi or the Faddeeva function w(z), the Dawson function can be extended to the entire complex plane:[2]

${\displaystyle F(z)={{\sqrt {\pi }} \over 2}e^{-z^{2}}{\mathrm {erfi} }(z)={\frac {i{\sqrt {\pi }}}{2}}\left[e^{-z^{2}}-w(z)\right]}$,

which simplifies to

${\displaystyle D_{+}(x)=F(x)={\frac {\sqrt {\pi }}{2}}\operatorname {Im} [w(x)]}$
${\displaystyle D_{-}(x)=iF(-ix)=-{\frac {\sqrt {\pi }}{2}}\left[e^{x^{2}}-w(-ix)\right]}$

for real x.

For |x| near zero, F(x) ≈ x, and for |x| large, F(x) ≈ 1/(2x). More specifically, near the origin it has the series expansion

${\displaystyle F(x)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}\,2^{k}}{(2k+1)!!}}\,x^{2k+1}=x-{\frac {2}{3}}x^{3}+{\frac {4}{15}}x^{5}-\cdots }$

F(x) satisfies the differential equation

${\displaystyle {\frac {dF}{dx}}+2xF=1\,\!}$

with the initial condition F(0) = 0.

## References

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1. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
2. Mofreh R. Zaghloul and Ahmed N. Ali, "Algorithm 916: Computing the Faddeyeva and Voigt Functions," ACM Trans. Math. Soft. 38 (2), 15 (2011). Preprint available at arXiv:1106.0151.