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