# Cauchy residue theorem

Let $U\subset\mathbb{C}$ be a simply connected domain, and suppose $f$ is a complex valued function which is defined and analytic on all but finitely many points $a_{1},\dots,a_{m}$ of $U$. Let $C$ be a closed curve in $U$ which does not intersect any of the $a_{i}$. Then

 $\int_{C}f(z)\ dz=2\pi i\sum_{i=1}^{m}\eta(C,a_{i})\operatorname{Res}(f;a_{i}),$

where

 $\eta(C,a_{i}):=\frac{1}{2\pi i}\int_{C}\frac{dz}{z-a_{i}}$

is the winding number of $C$ about $a_{i}$, and $\operatorname{Res}(f;a_{i})$ denotes the residue of $f$ at $a_{i}$.

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

Title Cauchy residue theorem CauchyResidueTheorem 2013-03-22 12:04:58 2013-03-22 12:04:58 djao (24) djao (24) 10 djao (24) Theorem msc 30E20 Cauchy residue formula residue theorem Residue CauchyIntegralFormula CauchyIntegralTheorem