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 $\infty$ 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

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

References

↑ 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.

This article is issued from Wikipedia - version of the Wednesday, December 09, 2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.

Residue at infinity

Definition

See also

References