Fourier transform of Gaussian multiplied by log

Question

I was wondering if the Fourier transform of the function

$$ f(x) = \frac{1}{\sqrt{2\pi}\delta}e^{-\frac{1}{2\delta^2}(x-\bar{x})^2}\log |x| $$

with $\delta,\bar{x}>0$, can be determined in closed form.

Answer 1

I think I found a way to do this, but it is slightly convoluted. I first compute the Fourier transform of

$$ f_n(x)=\frac{1}{\sqrt{2\pi}\delta}e^{-\frac{1}{2\delta^2}(x-\bar{x})^2}x^n $$

It turns out that this can be done in terms of some hypergeometric functions. Let us call the Fourier transform of this quantity $\mathcal{F}_n(k)$. Then, I take a derivative with respect to $n$ and set $n=1$. The final result can be expressed in terms of derivatives of ${}_1F_1(a;b;z)$ with respect to $a$ with $a=b=1/2$, and these can in terms be expressed as functions of generalised hypergeometric functions $_{2}F_2(\{a,b\};\{c,d\};z)$.