TITLE: What is $\sum (x+\mathbb{Z})^{-2}$?
QUESTION [14 upvotes]: This is a simple question, but its been bugging me.  Define the function $\gamma$ on $\mathbb{R}\backslash \mathbb{Z}$ by
$$\gamma(x):=\sum_{i\in \mathbb{Z}}\frac{1}{(x+i)^2}$$
The sum converges absolutely because it behaves roughly like $\sum_{i>0}i^{-2}$.
Some quick facts:

Pretty much by construction, $\gamma$ is periodic with period $1$.  
As it approaches any integer from the left or right, it goes to positive infinity.
It is symmetric at every half integer; that is, $\gamma(n+1/2+x)=\gamma(n+1/2-x)$ for all $n\in \mathbb{Z}$ and $x\in \mathbb{R}$.


Can $\gamma$ be expressed in terms of more familiar (presumably trigonometric) functions?

My best guess is $\gamma(x)=\sin^{-2}(\pi x)$, but this is based more on what I hope it would be, rather than what it is.

REPLY [3 votes]: For the record, one can prove the product formula for the sine without complex analysis (and without the Gamma function), from which (as David Speyer noted) one can recover $\sum_{i \in \bf Z} (x+i)^{-2}$ as the second logarithmic derivative.  See this Stackexchange answer.
[Later I learned that the case $x=1/2$ recovers Wallis's original proof
of his product formula for $\pi$; perhaps this generalization too was known before the complex-analytic proof.]