Cotangent partial-fraction Fourier kernel (${\sum }_{n\in \mathbb{Z}}(z+n{)}^{-k}$)

For an integer $k\ge 2$ and $\mathrm{Im}z>0$, differentiating the partial-fraction expansion of $\pi \cot \pi z$ givesBoth differentiated series converge locally uniformly. This kernel computes the Fourier expansion of a modular form for holomorphic Eisenstein series.

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 $\mathrm{Im}w>0$, the geometric-series formula for the cotangent also gives ${\pi }^2{\csc }^2(\pi w)=-4{\pi }^2{\sum }_{r\ge 1}r{e}^{2\pi irw}$.