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 [math]\displaystyle{ \infty }[/math] is a point added to the local space [math]\displaystyle{ \mathbb C }[/math] in order to render it compact (in this case it is a one-point compactification). This space denoted [math]\displaystyle{ \hat{\mathbb C} }[/math] is isomorphic to the Riemann sphere. One can use the residue at infinity to calculate some integrals.

Definition

Given a holomorphic function f on an annulus [math]\displaystyle{ A(0, R, \infty) }[/math] (centered at 0, with inner radius [math]\displaystyle{ R }[/math] and infinite outer radius), the residue at infinity of the function f can be defined in terms of the usual residue as follows:

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

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

Note that [math]\displaystyle{ \forall r \gt R }[/math], we have

[math]\displaystyle{ \operatorname{Res}(f, \infty) = {-1\over 2\pi i}\int_{C(0, r)} f(z) \, dz }[/math]

Since, for holomorphic functions the sum of the residues at the isolated singularities plus the residue at infinity is zero, it can be expressed as:

[math]\displaystyle{ \operatorname{Res}(f(z), \infty) = -\sum_k \operatorname{Res}\left(f\left(z\right), a_k\right). }[/math]

Motivation

One might first guess that the definition of the residue of [math]\displaystyle{ f(z) }[/math] at infinity should just be the residue of [math]\displaystyle{ f(1/z) }[/math] at [math]\displaystyle{ z=0 }[/math]. However, the reason that we consider instead [math]\displaystyle{ -\frac{1}{z^2}f\left(\frac{1}{z}\right) }[/math] is that one does not take residues of functions, but of differential forms, i.e. the residue of [math]\displaystyle{ f(z)dz }[/math] at infinity is the residue of [math]\displaystyle{ f\left(\frac{1}{z}\right)d\left(\frac{1}{z}\right)=-\frac{1}{z^2}f\left(\frac{1}{z}\right)dz }[/math] at [math]\displaystyle{ z=0 }[/math].

See also

• Riemann sphere
• Algebraic variety
• Residue theorem

References

• 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.