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 is a point added to the local space in order to render it compact. This space noted is isomorphic to the Riemann sphere. One can use the residue at infinity to calculate some integrals.

Definition

Given a holomorphic function f on an annulus , the residue at infinity of the function f can be defined in terms of the usual residue as follows:
Thus, one can transfer the study of at infinity to the study of at the origin.
Note that, we have