Dawson Function

In mathematics, the Dawson function or Dawson integral (named for John M. Dawson) is either

,

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

.

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

It is closely related to the error function erf, as

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

in terms of the real error function, erf.

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

,

which simplifies to

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

with the initial condition F(0) = 0.

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