# Residue at infinity

In complex analysis, a branch of mathematics, the residue at infinity is a residue of a holomorphic function on an annulus having an infinite external radius. The infinity $\infty$ is a point added to the local space $\mathbb C$ in order to render it compact (in this case it is a one-point compactification). This space noted $\hat{\mathbb C}$ is isomorphic to the Riemann sphere.[1] One can use the residue at infinity to calculate some integrals.

## Definition

Given a holomorphic function f on an annulus $A(0, R, \infty)$ (centered at 0, with inner radius $R$ and infinite outer radius), the residue at infinity of the function f can be defined in terms of the usual residue as follows:

$\mathrm{Res}(f,\infty) = \mathrm{Res}\left( {-1\over z^2}f\left({1\over z}\right), 0 \right)$

Thus, one can transfer the study of $f(z)$ at infinity to the study of $f(1/z)$ at the origin.

Note that $\forall r > R$, we have

$\mathrm{Res}(f, \infty) = {-1\over 2\pi i}\int_{C(0, r)} f(z) \, dz$