Cotangent Fourier expansion ($\pi \cot (\pi z)=-\pi i-2\pi i{\sum }_{r\ge 1}{e}^{2\pi irz}$)

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