What is 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 ∞ {\displaystyle \infty } is a point added to the local space C {\displaystyle \mathbb {C} } in order to render it compact. This space denoted C ^ {\displaystyle {\hat {\mathbb {C} }}} is isomorphic to the Riemann sphere. One can use the residue at infinity to calculate some integrals.