Cauchy's Residue Theorem

Theorem
Let $U$ be a simply connected open subset of the complex plane $\C$.

Let $a_1, a_2, \dots, a_n$ be finitely many points of $U$.

Let $f: U \to \C$ be analytic in $U \setminus \left\{{a_1, a_2, \dots, a_n}\right\}$.

Let $L = \partial U$ be oriented counterclockwise.

Then:
 * $\displaystyle \oint_L f \left({z}\right) \, \mathrm d z = 2 \pi i \sum_{k \mathop = 1}^n \operatorname{Res} \left({a_k}\right)$

where $\operatorname{Res}$ is the residue of $f$ at a point.

Proof
Let $\left\{ {U_1, \dotsc, U_n}\right\}$ be a set of open sets such that $U_i \subset U$, $a_i \in U_i$, $a_i \notin U_j$ for $i \ne j$.

Let $U_i \cap U_i = \varnothing$ for all $i \ne j$.

By Existence of Laurent Series, around each $a_k$ there is an expansion:


 * $\displaystyle \sum_{j \mathop = -\infty}^\infty c_j \left({z - a_k}\right)^j$

convergent in $f$ in $U_k$.

Then :