Residue theorem

This article gives the statement, and possibly proof, of a basic fact in complex analysis.
View a complete list of basic facts in complex analysis

Statement

Suppose $U$ is an open subset of $\mathbb{C}$. Let $z_1,z_2,\dots ,z_r$ be points in $U$ and $f:U\setminus \{z_j{\}}_{1\le j\le r}\to \mathbb{C}$ be a holomorphic function. Let $c$ be a 0-homologous sum of loops in $U\setminus \{z_j{\}}_{1\le j\le r}$ such that $c$ is zero-homologous. Then, we have:

${\oint }_{c}f(z)dz=\sum _{j=1}^{r}n(c;z_j)\mathrm{r}\mathrm{e}\mathrm{s}(f;z_j)$

The residue $\mathrm{r}\mathrm{e}\mathrm{s}$ here denotes the coefficient of $1/(z-z_j)$ in the laurent expansion about $z_j$.

Related facts

• Winding number version of Cauchy integral formula is a special case of this, where the function has precisely one pole of order one. However, we typically use the winding number version of Cauchy integral formula to establish the residue theorem.