$\pi^2/\sin^2(\pi z)=\sum_{n\in\mathbb{Z}}(z-n)^{-2}$

Define \[ f(z) &= {\pi^2 \over \sin^2 \qty{\pi z} } \\ g(z) &= \sum_{n\in \ZZ} {1\over (z-n)^2} .\]

1. Show that \(f\) and \(g\) have the same singularities in \(\CC\).
2. Show that \(f\) and \(g\) have the same singular parts at each of their singularities.
3. Show that \(f, g\) each have period one and approach zero uniformly on \(0\leq x \leq 1\) as \(\abs{y}\to \infty\).
4. Conclude that \(f = g\).