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. 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 Motivation One might first guess that the definition of the residue of f(z) at infinity should just be the residue of f(1/z) at z=0. However, the reason that we consider instead -f(1/z)/z2 is that one does not take residues of functions, but of differential forms, i.e. the residue of f(z)dz at infinity is the residue of f(1/z)d(1/z)=-f(1/z)dz/z2 at z=0. 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. Complex analysis