residue at infinity

If in the Laurent expansion

f(z)=k=-ckzk (1)

of the function f, the coefficient cn is distinct from zero (n>0) and  cn+1=cn+2==0,  then there exists the numbers M and K such that

|z-nf(z)|<Malways when|z|>K.

In this case one says that is a pole of order n of the function f (cf. zeros and poles of rational function).

If there is no such positive integer n, (1) infinitely many positive powers of z, and one may say that is an essential singularityMathworldPlanetmath of f.

In both cases one can define for f the residue at infinity as

12iπCf(z)𝑑z=c-1, (2)

where the integral is taken along a closed contour C which goes once anticlockwise around the origin, i.e. once clockwise around the point  z= (see the Riemann sphereMathworldPlanetmath).

Then the usual form


of the residue theoremMathworldPlanetmath may be expressed as follows:

The sum of all residues of an analytic functionMathworldPlanetmath having only a finite number of points of singularity is equal to zero.


  • 1 Ernst Lindelöf: Le calcul des résidus et ses applications à la théorie des fonctions.  Gauthier-Villars, Paris (1905).
Title residue at infinity
Canonical name ResidueAtInfinity
Date of creation 2013-03-22 19:15:00
Last modified on 2013-03-22 19:15:00
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 9
Author pahio (2872)
Entry type Definition
Classification msc 30D30
Related topic Residue
Related topic RegularAtInfinity