Residue at infinity

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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.

Definition[edit]

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

See also[edit]

References[edit]

  1. ^ Michèle AUDIN, Analyse Complexe, cursus 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 analytique des fonctions d'une ou plusieurs varaiables 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.