I am new at this and I am trying to get a hang of complex contour integration.
I would like to use Cauchy's residue theorem to evaluate the following integral (with real values of w):
where k,y > 0 and are real.
I know that in order to solve the line integral in question, by using this method, I need to find the sum of the residues at the poles and multiply it by 2*pi*i. I also know that one can go about computing the residues at the poles in one of two ways; by doing grunt-work derivatives or by doing series expansion.
I would like to take the most simple path, so I want to expand this function into a series. I have been trying to follow the example here: https://en.wikipedia.org/wiki/Residue_theorem, but I am not sure how to expand this in a useful way, that will lead to me find the residues.
Could somebody point me in the right direction?
Thanks in advance.