Cauchy residue theorem

Let $U\subset ℂ$ be a simply connected domain, and suppose $f$ is a complex valued function which is defined and analytic on all but finitely many points $a_1,\mathrm{\dots },a_m$ of $U$. Let $C$ be a closed curve in $U$ which does not intersect any of the $a_i$. Then

$${\int }_{C}f(z)𝑑z=2\pi i\sum _{i=1}^m\eta (C,a_i)\mathrm{Res}(f;a_i),$$

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

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