Cotangent partial-fraction expansion ($\pi \cot \pi z=z^{-1}+{\sum }_{n\ge 1}2z/(z^2-n^2)$)

Away from integer poles,The series is locally uniformly convergent. Integrate $\pi \cot (\pi \zeta )/({\zeta }^2-z^2)$ around the square with sides $\mathrm{Re}\zeta ,\mathrm{Im}\zeta =\pm (N+1/2)$. The cotangent is uniformly bounded there, so the integral tends to zero. The residue theorem then gives $\pi \cot (\pi z)/z+{\sum }_{n\in \mathbb{Z}}(n^2-z^2{)}^{-1}=0$, which is the expansion. Differentiation generates higher integer-lattice pole sums. Setting $w=e^{2\pi iz}$ gives $\pi \cot \pi z=\pi i(w+1)/(w-1)$. On $0<|z|<1$, expansion of each paired summand gives $\pi \cot \pi z=z^{-1}-2{\sum }_{m\ge 0}\zeta (2m+2)z^{2m+1}$, relating this trigonometric function to the Riemann zeta function.

For $\mathrm{Im}z>0$, writing cotangent as $i(e^{2\pi iz}+1)/(e^{2\pi iz}-1)$ and expanding a geometric series gives this Fourier series. Differentiating the cotangent partial-fraction expansion and this Fourier series $k-1$ times yields