Cauchy residue theorem

Let U be a simply connected domain, and suppose f is a complex valued function which is defined and analytic on all but finitely many points a1,,am of U. Let C be a closed curveMathworldPlanetmath in U which does not intersect any of the ai. Then




is the winding number of C about ai, and Res(f;ai) denotes the residue of f at ai.

The Cauchy residue theorem generalizes both the Cauchy integral theorem (because analytic functions have no poles) and the Cauchy integral formulaPlanetmathPlanetmath (because f(x)/(x-a)n for analytic f has exactly one pole at x=a with residue Res(f(x)/(x-a)n,a)=f(n)(a)/n!).

Title Cauchy residue theorem
Canonical name CauchyResidueTheorem
Date of creation 2013-03-22 12:04:58
Last modified on 2013-03-22 12:04:58
Owner djao (24)
Last modified by djao (24)
Numerical id 10
Author djao (24)
Entry type Theorem
Classification msc 30E20
Synonym Cauchy residue formula
Synonym residue theorem
Related topic Residue
Related topic CauchyIntegralFormula
Related topic CauchyIntegralTheorem