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We have the function $$ f(z)=\frac{\pi \cot(\pi z)}{(u + z)^2} $$ I already found the residue at the pole when $z = -u$. However there are more poles when z is an integer. How do I go about finding the sum of these poles? I don't think it's too difficult but I'm probably overlooking something simple. Just a kick in the right direction should do it. Thanks

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As you found, poles exist at integer $z$. Assuming $u$ is not an integer (to avoid double poles), and let $n \in \mathrm Z$:

$$\operatorname*{Res}_{z = n}\, f(z) = \lim_{z \to n}\, (z-n)f(z) = \lim_{z \to n}\, \pi \frac{(z-n)}{\sin(\pi z)}\frac{\cos(\pi z)}{(u+z)^2} = \frac{1}{(n+z)^2}$$

by L'Hopital's rule.

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