Dawson function

From formulasearchengine
Jump to navigation Jump to search
The Dawson function, , around the origin
The Dawson function, , around the origin

In mathematics, the Dawson function or Dawson integral (named for H. G. Dawson[1]) 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 Faddeeva function w(z), the Dawson function can be extended to the entire complex plane:[2]

,

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.

References

  • {{#invoke:citation/CS1|citation

|CitationClass=citation }}

  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.

External links


Template:Mathapplied-stub