proof of Cauchy residue theorem

Being f holomorphic by Cauchy-Riemann equationsMathworldPlanetmath the differential formMathworldPlanetmath f(z)dz is closed. So by the lemma about closed differential forms on a simple connected domain we know that the integral Cf(z)𝑑z is equal to Cf(z)𝑑z if C is any curve which is homotopic to C. In particular we can consider a curve C which turns around the points aj along small circles and join these small circles with segments. Since the curve C follows each segment two times with opposite orientation it is enough to sum the integrals of f around the small circles.

So letting z=aj+ρeiθ be a parameterization of the curve around the point aj, we have dz=ρieiθdθ and hence


where ρ>0 is chosen so small that the balls Bρ(aj) are all disjoint and all contained in the domain U. So by linearity, it is enough to prove that for all j


Let now j be fixed and consider now the Laurent seriesMathworldPlanetmath for f in aj:


so that Res(f,aj)=c-1. We have


Notice now that if k=-1 we have


while for k-1 we have


Hence the result follows.

Title proof of Cauchy residue theorem
Canonical name ProofOfCauchyResidueTheorem
Date of creation 2013-03-22 13:42:04
Last modified on 2013-03-22 13:42:04
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 7
Author paolini (1187)
Entry type Proof
Classification msc 30E20