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:


All the help is greatly appreciated!

  • 3
    $\begingroup$ For the definite integral, have you tried the Cauchy's residue theorem? $\endgroup$ Mar 21, 2018 at 19:21
  • $\begingroup$ 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! $\endgroup$ Mar 21, 2018 at 20:38
  • $\begingroup$ 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.) $\endgroup$ Mar 21, 2018 at 20:59


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