12. Theorem 2.9.5 -

Let $U \subseteq C$ be open, and $γ$ a closed $1$ -chain (a $1 - c y c l e$ ) in $U$ which is homologous to zero in $U$ . Let $z_{1}, \dots, z_{m}$ be a finite collection of points in $U$ , and $γ_{1}, \dots, γ_{m}$ "small" circles around $z_{1}, \dots, z_{m}$ oriented counterclockwise. Set $k_{j} = n_{γ} (z_{i})$ . Then we have that

γ \sim \sum_{j = 1}^{m} k_{j} γ_{j}

and furthermore for any $f$ holomorphic in $U - {z_{1}, \dots, z_{m}}$ , we have that
:?:

\oint_{γ} f d z = \sum_{j = 1}^{m} k_{j} \oint f d z