# Dawson function

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

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

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

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

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

$D_{+}(x)={\frac {1}{2}}\int _{0}^{\infty }e^{-t^{2}/4}\,\sin {(xt)}\,dt.$ It is closely related to the error function erf, as

$D_{+}(x)={{\sqrt {\pi }} \over 2}e^{-x^{2}}{\mathrm {erfi} }(x)=-{i{\sqrt {\pi }} \over 2}e^{-x^{2}}{\mathrm {erf} }(ix)$ where erfi is the imaginary error function, erfi(x) = −i erf(ix). Similarly,

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

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

which simplifies to

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

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

${\frac {dF}{dx}}+2xF=1\,\!$ with the initial condition F(0) = 0.