Dawson function

In mathematics, the Dawson function or Dawson integral[1] (named after H. G. Dawson[2]) is the one-sided Fourier–Laplace sine transform of the Gaussian function.

The Dawson function is defined as either:

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).

in terms of the real error function, erf.

In terms of either erfi or the Faddeeva function

the Dawson function can be extended to the entire complex plane:[3]

More specifically, near the origin it has the series expansion

satisfies the differential equation

Inflection points follow for

(Apart from the trivial inflection point at

) The Hilbert transform of the Gaussian is defined as

\int _{-\infty }^{\infty }{\frac {e^{-x^{2}}}{y-x}}\,dx}

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

{\displaystyle {1 \over u}=\int _{0}^{\infty }dk\,\sin ku=\int _{0}^{\infty }dk\,\operatorname {Im} e^{iku}.}

and complete the square with respect to

We complete the square with respect 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

is the Dawson function as defined above.

is also related to the Dawson function.

We see this with the technique of differentiating inside the integral sign.

\int _{-\infty }^{\infty }{\frac {x^{2n}e^{-x^{2}}}{y-x}}\,dx.}

\int _{-\infty }^{\infty }{e^{-ax^{2}} \over y-x}\,dx.}

\int _{-\infty }^{\infty }{\frac {x^{2n}e^{-ax^{2}}}{y-x}}\,dx.}

The derivatives are performed first, then the result evaluated at

can be calculated using the recurrence relation (for