Integrating this double integral

[Baggio]

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

[tex]\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'[/tex]

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

[tex]e^{-2k|\tau|}[/tex]

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
[tex]\int \frac{k}{\pi} \frac{1}{k^2 + \omega^2} e^{-i\omega \tau} d\omega[/tex]
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.

 

[tacman]

Integrating double integrals can definitely be challenging, especially with complex expressions like the one you have provided. One approach you can try is to use the substitution method. This involves substituting new variables for the original ones in the integral and then solving for the new limits of integration. In this case, you can try substituting u = k^2 + \omega'^2 and v = k^2 + \omega^2. This will allow you to rewrite the integral in terms of u and v, making it easier to integrate. Additionally, using the properties of complex numbers, you can simplify the exponential terms in the integrand. Overall, this method may give you a better chance of getting the desired result of e^{-2k|\tau|}. I hope this helps and good luck with your integration!