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

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