In mathematics, the Dawson function or Dawson integral (named for H. G. Dawson)
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 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) satisfies the differential equation
with the initial condition F(0) = 0.