Polar Fourier transform of derivatives

[hunt_mat]

TL;DR Summary

Computing Fourier transforms is simple in Cartesian co-ordinates but how do you do it for polar co-ordinates?

The 2D Fourier transform is given by: $$\hat{f}(k,l)=\int_{\mathbb{R}^{2}}f(x,y)e^{-ikx-ily}dxdy$$
In terms of polar co-ordinates: $$\hat{f}(\rho,\phi)=\int_{0}^{\infty}\int_{-\pi}^{\pi}rf(r,\theta)e^{-ir\rho\cos(\theta-\phi)}drd\theta$$

For Fourier transforms in cartesian co-ordinates, relating the Fourier transform of a derivative of a function to the Fourier transform of the function. However, what happens with the polar Fourier transform? I've done a simple calculation for the $r-$derivative and I get:
$$\widehat{\frac{\partial f}{\partial r}}=-\rho\sum_{n\in\mathbb{Z}}\int_{0}^{\infty}rf_{n}(r)J_{n-1}(r\rho)dr$$
This is after I expanded $f(r,\theta)$ as a Fourier series:$$f(r,\theta)=\sum_{n\in\mathbb{Z}}f_{n}(r)e^{in\theta}$$
I feel a little lost at this point. Can anyone suggest anything?

 

[DaveE]

TL;DR Summary: Computing Fourier transforms is simple in Cartesian co-ordinates but how do you do it for polar co-ordinates?

The 2D Fourier transform is given by: [tex]\hat{f}(k,l)=\int_{\mathbb{R}^{2}}f(x,y)e^{-ikx-ily}dxdy[\tex]

Did you mean to type this?
The 2D Fourier transform is given by: $\hat{f}(k,l)=\int_{\mathbb{R}^{2}}f(x,y)e^{-ikx-ily}dxdy$

 

[hunt_mat]

Did you mean to type this?
The 2D Fourier transform is given by: $\hat{f}(k,l)=\int_{\mathbb{R}^{2}}f(x,y)e^{-ikx-ily}dxdy$

I was correcting my memory on how to include LaTeX here, you'll see that it's corrected now.

 

[pasmith]

Search for "Hankel transform". The Wikipedia article doesn't give an expression for the transform of a derivative, but differentiating the inverse transform and using Bessel function identities should work.

 

[hunt_mat]

I had a feeling that this would be the case. A lot of my undergraduate was pure maths, mainly analysis and mathematical physics. Solving PDEs was something I had to teach myself about.

This is about a water wave problem, looking for the free surface from a vorticity distribution. The azimuthal component will be the most important.