Residue theorem

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:

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

The residue $res$ 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.