How to compute the 2D inverse Fourier transform?

[bgturk]

Homework Statement

The problem is to obtain the inverse Fourier transform of the following 2D functions

$$F(\mathbf{k})=\frac{k_{x}k_{y}}{k^{2}}$$

Homework Equations

The relevant equations are the 2d Fourier transform formulas described http://fourier.eng.hmc.edu/e101/lectures/Image_Processing/node6.html" .

The Attempt at a Solution

$$\int d^{2}\mathbf{k}\,\frac{k_{x}k_{y}}{k^{2}}e^{i\mathbf{k}\cdot\mathbf{r}}&=&\int_{-\infty}^{+\infty}dp\int_{-\infty}^{+\infty}dq\,\frac{p q}{p^{2}+q^{2}}e^{ipx+iqy}$$

How would you proceed with the evaluation of this integral? I need some guidance on how to compute integrals like the above.

 

[lanedance]

could you try polar co-ords?