Dawson function

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The Dawson function, D + (x), around the origin
The Dawson function, D + (x), around the origin
The Dawson function, D − (x), around the origin
The Dawson function, D − (x), around the origin

In mathematics, the Dawson function is

F(x) = e^{-x^2} \int_0^x e^{t^2}\,dt.

The notation D(x) is also in use. The Dawson function is also called the Dawson integral.

The Dawson function is closely related to the error function erf, as

F(x) = {\sqrt{\pi} \over 2} e^{-x^2} \mathrm{erfi} (x) = - {i \sqrt{\pi} \over 2} e^{-x^2} \mathrm{erf} (ix)

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

For |x| near zero, F(x)\approx x , and for |x| large, F(x)\approx \frac{1}{2x} .

F(x) satisfies the differential equation

\frac{dF}{dx} + 2xF=1

with the initial condition F(0)=0.

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