Residue theorem

Statement

Let $U \subseteq \mathbb{C}$ be a simply connected open set. This means that $U$ is connected, and each closed path in $U$ can be continuously deformed in $U$ to a constant path. Let $a_1, \dots, a_n$ be distinct points of $U$, and let $f$ be holomorphic on $U \setminus \{a_1, \dots, a_n\}$. Let $\gamma$ be a closed, continuous and piecewise continuously differentiable path in $U \setminus \{a_1, \dots, a_n\}$. Then $$\oint_\gamma f(z) \, dz = 2 \pi i \sum_{k=1}^{n} n(\gamma, a_k) \operatorname{Res}(f, a_k),$$ where $n(\gamma, a_k)$ is the winding number of $\gamma$ around $a_k$ and $\operatorname{Res}(f, a_k)$ is the residue of $f$ at $a_k$.