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An oriented piecewise differentiable path in the complex plane, used as the domain of a contour integral. Its orientation controls the signs of residues in the residue theorem.

Contour shifting deforms a complex integration path through a region where the integrand is meromorphic. The residue theorem says that the change equals $2\pi i$ times the sum of residues of the poles crossed.

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 ${\pi }^2{\csc }^2(\pi w)=-4{\pi }^2{\sum }_{r\ge 1}r{e}^{2\pi irw}$.

For a pole off the real axis, substitution $s=2t$ into the Faddeeva function integral gives $G(a)=i\pi w(a/2)$ when $\mathrm{Im}a>0$, and $G(a)=-i\pi w(-a/2)$ when $\mathrm{Im}a<0$. The difference of boundary values is the residue theorem jump $2\pi i{e}^{-a^2/4}$. In particular, for a lower-half-plane pole, $G(a)+2\pi i{e}^{-a^2/4}=i\pi w(a/2)$, which packages a crossed residue and a nearby saddle into one smooth expression.

Let $g$ be a meromorphic function with simple poles ${\alpha }_{+}$ above $c$ and ${\alpha }_{-}$ below $-c$, and $zg(z)\to 0$ uniformly at infinity in $\mathrm{Im}z\ge -c$. The left-to-right line integrals $H(\lambda )=(2\pi i{)}^{-1}{\int }_{\mathbb{R}-ic}g(z)/(z-\lambda )dz$ and $J(\lambda )=-(2\pi i{)}^{-1}{\int }_{\mathbb{R}+ic}g(z)/(z-\lambda )dz$ satisfy $H=g-r_{+}/(\lambda -{\alpha }_{+})$ above $-c$ and $J=r_{+}/(\lambda -{\alpha }_{+})$ below $c$, where $r_{+}$ is the upper residue. Upper semicircle closure proves both formulas by the residue theorem; subtracting the principal part makes $H$ holomorphic at ${\alpha }_{+}$ too. Consequently $H+J=g$ on the intervening strip. No lower-half-plane decay is needed.

A contour encircling a branch cut travels on its two sides and joins them by small and large circles. It converts the phase change of a complex power into a real integral through the residue theorem. The two circular contributions must be bounded before taking their limiting radii.