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:

\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

References

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.

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[1] Michèle Audin, Analyse Complexe, lecture notes of the University of Strasbourg available on the web, pp. 70–72

Residue at infinity

Definition

See also

References