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.



What is $\sum (x+\mathbb{Z})^{-2}$?
Use residues! For an entertaining narrative with the correct answer, see [here](https://mathoverflow.net/questions/8741/justifying-a-theory-by-a-seemingly-unrelated-example/8752#8752). For a derivation, see a complex analysis text.


### Added


Here's a link to an outline of the residue method of solving this: [Conway page 122](http://books.google.com/books?id=9LtfZr1snG0C&lpg=PP1&dq=complex%2520analysis%2520conway&pg=PA122#v=onepage&q=&f=true).


