I' m looking for a way to relate the Fourier trasform in the form
$$ \tilde F[\vec q] = \int d^2x \, e^{i \vec q \cdot \vec x} f\left[ \sqrt{x^2 + d^2} \right], $$
and the fourier transform
$$F'[\vec q] = \int d^2x \, e^{i \vec q \cdot \vec x} f\left[ \vert x \vert \right], $$
by trasnforming the first integral with the change of variables $ \vec x \rightarrow \vec x - i \vec d$, with $\vec d \parallel \vec q$. This problem is the same as that of valuating the Fourier transform of a 2D Coulomb potential $V \sim 1/r$ and $V \sim 1/\sqrt{r^2 + d^2}$.
As a result, I expect to see something like:
$$\tilde F[\vec q] = e ^ {- q d} F'[\vec q],$$
but the procedure would require to go into the complex plane and doesn' t seem intuitive how to get the expected result. Is there any hint to get there? Or is there a more appropriate way to relate the two?