Dawson function

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The Dawson function, $F(x) = D_+(x)$, around the origin
The Dawson function, $D_-(x)$, around the origin

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

$D_+(x) = \frac12 \int_0^\infty e^{-t^2/4}\,\sin{(xt)}\,dt.$

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:[2]

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

which simplifies to

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

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.

References

1. ^ Dawson, H. G. (1897). "On the Numerical Value of $\int_0^h \exp(-x^2) dx$". Proceedings of the London Mathematical Society. s1-29 (1): 519–522. doi:10.1112/plms/s1-29.1.519.
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.