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 denoted $\hat$ is isomorphic to the Riemann sphere.Michèle Audin, ''Analyse Complexe'', lecture notes of the University of Strasbour
available on the web
pp. 70–72 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:

:$\operatorname(f,\infty) = -\operatorname\left( f\left(\right), 0 \right)$

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

:$\operatorname(f, \infty) = \int_ f(z) \, dz$

Motivation

One might first guess that the definition of the residue of $f(z)$ at infinity should just be the residue of $f(1/z)$ at $z=0$. However, the reason that we consider instead $-\fracf(\frac)$ is that one does not take residues of ''functions'', but of ''differential forms'', i.e. the residue of $f(z)dz$ at infinity is the residue of $f(\frac)d(\frac)=-\fracf(\frac)dz$ at $z=0$.

See also

* Riemann sphere
* Algebraic variety
* Residue theorem

References

* Murray R. Spiegel, ''Variables complexes'', Schaum,
* 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.

Complex analysis