= Cosecant partial-fraction identity
{title2=$\sum_{n\in\mathbb Z}(w+n)^{-2}=\pi^2\csc^2(\pi w)$}
For $w\notin\mathbb Z$, integrate $\pi\cot(\pi\zeta)/(\zeta-w)^2$ around large squares whose sides have half-integer coordinates. The <cotangent> is uniformly bounded on these sides and the integral tends to zero. The <residue theorem> gives residues $(n-w)^{-2}$ at the integers and $-\pi^2\csc^2(\pi w)$ at $w$, proving the identity. For $\operatorname{Im}w>0$, the geometric-series formula for the <cotangent> also gives $\pi^2\csc^2(\pi w)=-4\pi^2\sum_{r\geq1}r e^{2\pi irw}$.
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