I am trying to compute the 2-d Fourier transform: $$I=\int d^2\vec{r} \frac{e^{i \vec{q} \vec{r}}}{(1+r^2)^{3/2}}$$ In polar coordinates the integral looks like:
$$I=\int^{2\pi}_0d \theta \int^{\infty}_0 dr \frac{r}{(1+r^2)^{3/2}}e^{i q r \cos\theta}$$
I have tried to integrate over the angle first to get a Bessel function, but when I expand it as a series, the integral over $r$ do not converge.
Mathematica gives a simple answer, that is why I am curious on how to derive it.