Dawson function

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The Dawson function, , around the origin
The Dawson function, , around the origin

In mathematics, the Dawson function or Dawson integral (named after 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

,

while for large x it has the asymptotic expansion

,

where n!! is the double factorial.

F(x) satisfies the differential equation

with the initial condition F(0) = 0. Consequently, it has extrema for

,

resulting in x = ±0.92413887… (OEISA133841), F(x) = ±0.54104422… (OEISA133842).

Inflection points follow for

,

resulting in x = ±1.50197526… (OEISA133843), F(x) = ±0.42768661… (OEISA245262). (Apart from the trivial inflection point at x = 0, F(x) = 0.)

Relation to Hilbert transform of Gaussian[edit]

The Hilbert Transform of the Gaussian is defined as

P.V. denotes the Cauchy principal value, and we restrict ourselves to real . can be related to the Dawson function as follows. Inside a principal value integral, we can treat as a generalized function or distribution, and use the Fourier representation

With , we use the exponential representation of and complete the square with respect to to find

We can shift the integral over to the real axis, and it gives . Thus

We complete the square with respect to and obtain

We change variables to :

The integral can be performed as a contour integral around a rectangle in the complex plane. Taking the imaginary part of the result gives

where is the Dawson function as defined above.

The Hilbert transform of is also related to the Dawson function. We see this with the technique of differentiating inside the integral sign. Let

Introduce

The nth derivative is

We thus find

The derivatives are performed first, then the result evaluated at . A change of variable also gives . Since , we can write where and are polynomials. For example, . Alternatively, can be calculated using the recurrence relation (for )

.

References[edit]

  1. ^ Dawson, H. G. (1897). "On the Numerical Value of ". 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.

External links[edit]