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