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:

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

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

• Riemann sphere
• Algebraic variety
• Residue theorem

References

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.