The Residue Theorem

# The Residue Theorem

Theorem 1 (The Residue Theorem): Let $U \subseteq \mathbb{C}$ be a simply connected and open set and let $\{ a_1, a_2, ..., a_n \} \subset U$ be a finite collection of points in $U$. Let $f$ be a complex function that is holomorphic on $U \setminus \{ a_1, a_2, ..., a_n \}$ and let $\gamma$ be a closed, piecewise smooth curve in $U$ enclosing (but not passing through) the points $\{ a_1, a_2, ..., a_n \}$. Then $\displaystyle{\oint_{\gamma} f(z) \: dz = 2\pi i \sum_{k=1}^{n} n(\gamma, a_k) \mathrm{Res}(f, a_k)}$. |

*In particular, if $\gamma$ is a simple closed piecewise smooth positively oriented curve, then $n(\gamma, a_k) = 1$ for each $k$ and $\displaystyle{\oint_{\gamma} f(z) \: dz = 2\pi i \sum_{k=1}^{n} \mathrm{Res}(f, a_k)}$.*