TITLE: How many ways to calculate: $\sum_{n=-\infty}^{+\infty}\frac{1}{(u+n)^2}$ where $u \not \in \Bbb{Z}$
QUESTION [10 upvotes]: Today I have encounter a series:
$$\sum_{n=-\infty}^{+\infty}\frac{1}{(u+n)^2}=\frac{\pi^2}{(\sin \pi u)^2}$$
where $u \not \in \Bbb{Z}$
. I have known a method to computer it (by Residue formula):
$$\int_{|z|=N+1/2}\frac{\pi \cot \pi z}{(u+z)^2}\text{dz}=-\frac{\pi^2}{\sin^2 \pi u}+\sum_{k=-N}^{N}\frac{1}{(u+k)^2}$$
where $-u$ is a pole of order 2. and $n \in \Bbb{Z}$ and $|n| \leq N$ are poles of order 1. because $$\left|\int \frac{\pi \cot \pi z}{(u+z)^2}\text{dz}\right|\leq \int_0^{2\pi}\pi\left|1+\frac{2}{e^{i2\pi z}-1}\right|\frac{1}{|((N+1/2)e^{i\varphi}+u)^2|}\text{d}\varphi \rightarrow 0$$ as $N \rightarrow +\infty$
Dose there exists some other way to computer it?

REPLY [3 votes]: Let $f(z) =\displaystyle \sum_{n\in \mathbb{Z}} \frac{1}{(n+z)^2}$ and $g(z) = \dfrac{\pi^2}{\sin^2(\pi z)}.$ Both functions are $1$ periodic, so it suffices to understand their behavior in the strip $\mathfrak{R}(z)\in [0,1).$ It is straightforward to verify that both these functions have a singularity only at the origin and are bounded on the strip excluding any neighborhood of the origin. 
Now look near $z=0.$ We have $f(z) =\displaystyle \frac{1}{z^2} + \sum_{n\in\mathbb{Z}\setminus\{0\}} \frac{1}{(z+n)^2}.$ By the power series expansion of sine and geometric series, we have $$ g(z) = \frac{\pi^2}{\pi^2 z^2 - (\pi^4/3)z^4 + \mathcal{O}(z^6) } = \frac{1}{z^2} \left( \frac{1}{1- (\pi^4/3) z^2 + \mathcal{O}(z^4) } \right) = \frac{1}{z^2} \left( 1 + (\pi^2/3) z^2 + \mathcal{O}(z^4) \right) = \frac{1}{z^2} +\pi^2/3 + \mathcal{O}(z^2).$$ 
So in fact $f(z)-g(z)$ is bounded and analytic on the strip, and hence the whole complex plane. By Liouville's theorem, $f(z)-g(z) = f(0)-g(0)=0$ so $f(z) = g(z).$