What is the Fourier transform of $\displaystyle \operatorname{Erf}\left[a+bx^{2}\right] $?
I need this in order to evaluate
$$ \int_{-\infty}^{\infty}e^{-\beta x^{2}}Erf\left[a+bx^{2}\right]dx $$
using Parseval's theorem:
$$\int_{-\infty}^{\infty} dx \, f(x) g(x) = \frac1{2 \pi} \int_{-\infty}^{\infty} dk \, F(k) G(k)$$