Residue at infinity explained

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

inf ty 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 denoted \hat{ C} is isomorphic to the Riemann sphere.^[1] One can use the residue at infinity to calculate some integrals.

Definition

A( 0 , R, inf ty) (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:\operatorname{Res}(f, inf ty) = - \operatorname{Res}\left( { 1 \over z ^{2}f\left({1\over} z}\right), 0 \right)

Thus, one can transfer the study of

f(z) at infinity to the study of f( 1 /z) at the origin.

Note that

\forall r > R , we have\operatorname{Res}(f, inf ty) = { -1 \over 2 \pi i}\int _{C(0,} f(z) dz 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:\operatorname{Res}(f(z), inf ty) = - \sum _{k} \operatorname{Res}\left(f\left(z\right), a _{k\right).}

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 - (1)/(f\left( z ^{2}) (1)/(z) \right) 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\left( (1)/(\right)d\left( z) (1)/(\right) =- z) (1)/(f\left( z ^{2}) (1)/(z) \right)dz at z =0 .

See also

• Riemann sphere
• Algebraic variety
• Residue theorem

References

1. Michèle Audin, Analyse Complexe, lecture notes of the University of Strasbourg available on the web, pp. 70–72

• 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.