Cauchy residue theorem

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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 $\operatorname{Res}(f(x)/(x-a)^{n},a)=f^{{(n)}}(a)/n!)$ .

Synonym:

Cauchy residue formula, residue theorem

Type of Math Object:

Theorem

Major Section:

Reference

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