# Solution of the Yukawa potential Fourier representation integral

I've been trying to get my head around solving the integral, which is the Fourier Transform representation of the Yukawa potential in k-space. Anyhow, the specifics don't matter, I am simply stuck on the last step of integration. This is the presented solution on Wikipedia:

The step, which troubles me is the following:

$\frac{1}{4\pi^2ir}\int_{-\infty}^{\infty}dk\frac{ke^{ikr}}{(k+im)(k-im)}=\frac{1}{4\pi^2ir}2i\pi\frac{im}{2im}e^{-mr}$

All the help is greatly appreciated!

• For the definite integral, have you tried the Cauchy's residue theorem? Mar 21, 2018 at 19:21
• No, I haven't. Indeed, I admit, I was not familiar with it up to this point. I'll look into the theory more fully and give it a go. It seems to provide the answer on the first glance. Much appreciated! Mar 21, 2018 at 20:38
• If $r>0$, it seems that we may consider the contor: 1. The straight line from $(-R,0)$ to $(R,0)$; 2. The upper semi-circle centered at $(0,0)$ with radius $R$, from $(R,0)$ to $(-R,0)$ anti-clockwisely. (I have no time to work out the details.) Mar 21, 2018 at 20:59