# Finding the simple poles and getting the sum of residues

1. Jan 13, 2012

### HACR

1. The problem statement, all variables and given/known data
$$C_{N}$$ denotes the positively oriented boundary of the square whose edges lie along the lines $$x=+/- (N+1/2\pi), y=+/- (N+1/2\pi)$$.

Then show that [tex]\int_{C_{N}} \frac{dz}{z^{2}sin(z))=2\pi i[\frac{1}{6}+2\sum_{n=1}^{N}\frac{(-1)^n}{n^2{\pi}^2}]
2. Relevant equations

3. The attempt at a solution
So the problem boils down to finding the simple poles and getting the sum of residues. the first one occurs at z=0. But for this one, this is a double pole for z, and infinitely many zeros for sin(z) at infinity (pole at zero). So then how to determine the residue of the first pole?