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 ${\hat {\mathbb {C} }}$ 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 $A(0,R,\infty )$ (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

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

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:

$\operatorname {Res} (f(z),\infty )=-\sum _{k}\operatorname {Res} \left(f\left(z\right),a_{k}\right).$

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 $-{\frac {1}{z^{2}}}f\left({\frac {1}{z}}\right)$ 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 $f\left({\frac {1}{z}}\right)d\left({\frac {1}{z}}\right)=-{\frac {1}{z^{2}}}f\left({\frac {1}{z}}\right)dz$ at $z=0$.

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.