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 (in this case it is a one-point compactification). This space denoted is isomorphic to the Riemann sphere.Michèle Audin, Analyse Complexe, lecture notes of the University of Strasbourg available on the web , pp. 70–72 One can use the residue at infinity to calculate some integrals.
Definition Given a holomorphic function f on an annulus (centered at 0, with inner radius and infinite outer radius), 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
Since, for holomorphic functions the sum of the residues at the isolated singularities plus the residue at infinity is zero, it can be expressed as:
Motivation
One might first guess that the definition of the residue of at infinity should just be the residue of at . However, the reason that we consider instead is that one does not take residues of functions, but of differential forms, i.e. the residue of at infinity is the residue of at .
See also Riemann sphere Algebraic variety Residue theorem
References
Murray R. Spiegel, Variables complexes, Schaum, Henri Cartan, Théorie élémentaire des fonctions analytiques d'une ou plusieurs variables complexes, Hermann, 1961 Mark J. Ablowitz & Athanassios S. Fokas, Complex Variables: Introduction and Applications (Second Edition), 2003, , P211-212.