Homework Help: How to compute the 2D inverse Fourier transform?

1. May 7, 2010

bgturk

1. The problem statement, all variables and given/known data

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

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

2. Relevant equations

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

3. 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.

Last edited by a moderator: May 4, 2017
2. May 8, 2010

lanedance

could you try polar co-ords?