Dawson function

Summary

The Dawson function is $F(x) = e^{-x^2} \int_0^x e^{t^2}\, dt$. It is a scaled form of the imaginary error function: $F(x) = \frac{\sqrt{\pi}}{2} e^{-x^2} \operatorname{erfi} x$, where $\operatorname{erfi} x = -i \operatorname{erf}(ix)$ and $\operatorname{erf}$ is the error function. It is the imaginary part of the Faddeeva function on the real line, up to the factor $\frac{2}{\sqrt{\pi}}$. It appears in plasma physics, in spectroscopy and in the Hilbert transform of the Gaussian.