Does anyone know the Fourier transform of
$\Large\frac{\text{erf}(\omega x)}{x}$?
I think it should be something like $\frac{4\pi}{k^2}\exp{(-k^2/4\omega^2)}$.
Is this right? How can one go about deriving this? Any hints are much appreciated.
Thank you in advance!
I think you have an error in your question: $\omega$ is typically used for the transform variable, and hence certainly shouldn't be in the untransformed function.
Anyway, the Fourier transform of ${\rm{Erf}(x) \over x}$ (of your title) is:
$$\frac{\Gamma \left(0,\frac{\omega ^2}{4}\right)}{\sqrt{2 \pi }}$$