Residue at infinity

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 noted is isomorphic to the Riemann sphere.[1] One can use the residue at infinity to calculate some integrals.

DefinitionEdit

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

 


MotivationEdit

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 alsoEdit

ReferencesEdit

  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, 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.