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 C in order to render it compact (in this case it is a one-point compactification). This space noted C ^ is isomorphic to the Riemann sphere. One can use the residue at infinity to calculate some integrals.
Given a holomorphic function f on an annulus A ( 0 , R , ∞ ) (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:
R e s ( f , ∞ ) = − R e s ( 1 z 2 f ( 1 z ) , 0 ) Thus, one can transfer the study of f ( z ) at infinity to the study of f ( 1 / z ) at the origin.
Note that ∀ r > R , we have
R e s ( f , ∞ ) = − 1 2 π i ∫ C ( 0 , r ) f ( z ) d z 