I am trying the following problem from Fisher's Complex Variables book:
If $f$ is analytic on a plane except at poles $\gamma_1, \cdots \gamma_N$ and none of them are integers and $\lim\limits_{z\to\infty} |z(f(z)|= 0$. Show that $\sum_{-\infty}^{+\infty} f(n)= -\sum_{j=1}^{+\infty}\operatorname{Res} (fC; \gamma_j)$ and $C(z)=\pi \cot(\pi z)$
Based on the hint provided I was able to prove that the residue was $f$. However I am not sure how to attempt this problem further.
Hint: $\pi \cot(\pi z) = \frac{1}{z}+2z\sum_{n=1}^\infty\frac{1}{z^2-n^2}$.
Also, notice that $|zf(z)|\rightarrow 0$ gives you nice convergence for $\oint_R f(z)C(z)dz$ when you take $R\rightarrow\infty$, where the contour integral is on a disk of radius $R$ about the origin.