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.^[1] One can use the residue at infinity to calculate some integrals.

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:

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 .

• Riemann sphere
• Algebraic variety
• Residue theorem

• Murray R. Spiegel, Variables complexes, Schaum, ISBN 2-7042-0020-3
• 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, ISBN 978-0-521-53429-1, P211-212.