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 ${\displaystyle \infty }$ is a point added to the local space ${\displaystyle \mathbb {C} }$ in order to render it compact (in this case it is a one-point compactification). This space noted ${\displaystyle {\hat {\mathbb {C} }}}$ 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 ${\displaystyle A(0,R,\infty )}$ (centered at 0, with inner radius ${\displaystyle R}$ and infinite outer radius), the residue at infinity of the function f can be defined in terms of the usual residue as follows:

${\displaystyle \operatorname {Res} (f,\infty )=-\operatorname {Res} \left({1 \over z^{2}}f\left({1 \over z}\right),0\right)}$

Thus, one can transfer the study of ${\displaystyle f(z)}$ at infinity to the study of ${\displaystyle f(1/z)}$ at the origin.

Note that ${\displaystyle \forall r>R}$, we have

${\displaystyle \operatorname {Res} (f,\infty )={-1 \over 2\pi i}\int _{C(0,r)}f(z)\,dz}$

See also

• Riemann sphere
• Algebraic variety
• Residue theorem

References

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.