[Math] Fourier transform of a 2-D Gaussian on a ring

Question

I need some help obtaining the 2-D Fourier transform of the following function:

$$f(r)=e^{-\frac{-2(r-a)^{2}}{w^{2}}}$$

Where $r$ is the polar radius, $a$ and $w$ are positive. So this describes a radially symmetric Gaussian on a ring of radius $a$. I tried solving it for $a=0$ and applying a simple translation, but this doesn't seem to work as it does in the 1-D case, since this is really a Hankel transform due to the radial symmetry. The definition of a (edit: 1-D) Fourier transform which I am using is:

$$\hat{f}(k)=\int_{-\infty}^{\infty}f(x)\ e^{-ixk}\, dx$$

Does anyone have any idea how to approach this?

Edit: Just to confirm, the integral I need seems to be $$\int_{0}^{\infty}f(r)\, J_{0}(kr)rdr$$

Answer 1

Please check page 7.7.3 of ERDELYI_Higher Transcendental Functions_Volume 2.

The integral closest to the one that you need is: