Residue at infinity

Definition

Given a holomorphic function f on an annulus A(0, R, \infty) (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,\infty) = -\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, \infty) = {-1\over 2\pi i}\int_{C(0, r)} 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), \infty) = -\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 -\frac{1}{z^2}f\left(\frac{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(\frac{1}{z}\right)d\left(\frac{1}{z}\right)=-\frac{1}{z^2}f\left(\frac{1}{z}\right)dz at z=0.

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.

References

1. Michèle Audin, ''Analyse Complexe'', lecture notes of the University of Strasbourg [http://www-irma.u-strasbg.fr/~maudin/analysecomp.pdf available on the web] {{Webarchive. link. (2012-01-25 , pp. 70–72)