Integrating this double integral

[Baggio]

Hi, I am having some difficulties integrating the following expression..

\int\int\left(\frac{k}{\pi}\right)^2 \frac{1}{k^2+\omega'^2}\frac{1}{k^2+\omega^2}e^{-i\omega'\tau}e^{i\omega\tau}d\omega d\omega'

I've tried by part but it doesn't look like it's going to give me the right answer which is

e^{-2k|\tau|}

Omega and Omega primed can be integrated from -infinity to infinity ... if that helps

Any ideas?

 

[Xevarion]

Well this is just the product of two integrals like
\int \frac{k}{\pi} \frac{1}{k^2 + \omega^2} e^{-i\omega \tau} d\omega
so all you have to do is integrate one of those.

As for how to do it: do you know any complex analysis? The easiest way to compute this integral is to use the residue theorem. If you don't know the residue theorem, you might want to learn some complex; if you do, this should be enough of a hint already.