TITLE: Fourier transform of the error function, erf (x)
QUESTION [11 upvotes]: I define $\text{erf}(x):=\frac{2}{\sqrt{\pi}}\int_0^xe^{-\xi^2}d\xi$. 
What is its Fourier transform (unitary, ordinary frequency)? That is, simplify
$$\frac{2}{\sqrt{\pi}}\int_{-\infty}^\infty\int_0^xe^{-\xi^2}e^{-i2\pi fx}d\xi dx $$
The answer does not seem to be easily accessible on the internet (until now).

REPLY [2 votes]: This result is reasonably well known; it is included in the Bateman MS (vol. I) and cited in Mehta's "Random Matrices" as A.31.1. A direct proof exploits the fact that erf is an odd function of its argument and a quick integration by parts.