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- Jan 26, 2012

- 995

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**Problem**: Suppose $u$ is not an integer. Prove that\[\sum_{n=-\infty}^{\infty}\frac{1}{(u+n)^2} = \frac{\pi^2}{(\sin \pi u)^2}\]

by integrating

\[f(z)=\frac{\pi\cot\pi z}{(u+z)^2}\]

over the circle $|z|=R_N=N+1/2$ ($N$ integral, $N\geq |u|$), adding the residues of $f$ inside the circle, and letting $N$ tend to infinity.

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