Residue at infinity

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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 \infty is a point added to the local space \mathbb C in order to render it compact (in this case it is a one-point compactification). This space noted  \hat{\mathbb C} 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  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:

 \mathrm{Res}(f,\infty) = \mathrm{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

 \mathrm{Res}(f, \infty) = {-1\over 2\pi i}\int_{C(0, r)} f(z) \, dz

See also[edit]

References[edit]

This article incorporates information from this version of the equivalent article on the French Wikipedia.
  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