Cotangent contour summation (${\mathrm{Res}}_{z=n}[\pi \cot (\pi z)h(z)]=h(n)$)

The cotangent kernel $\pi \cot (\pi z)$ has a simple pole of residue one at every integer. If a meromorphic $h$ is regular at the integers, these residues sample its integer values. Integrating around squares of half-side $N+1/2$ and applying the residue theorem converts a finite sum of samples into residues at the other poles. When the contour integral tends to zero and $h$ has only finitely many other poles, the limiting sum is the negative sum of those off-integer residues of $\pi \cot (\pi z)h(z)$. A uniform cotangent bound and sufficiently rapid decay of $h$ justify the vanishing contour term. At an integer where $h$ itself has a pole, use the full Laurent series to find the combined residue instead of substituting a divergent sample; the Basel problem calculation with $h(z)=z^{-2}$ illustrates this exception.