Let $G\subseteq \mathbb{C}$ be a domain, and let $S\subseteq G$ be a set without accumulation points in itself (I think we need that it has no accumulation points in $G$?). Let $f:G-S\to \mathbb{C}$ be holomorphic, and $\gamma :[t_0,t_1]\to G-S$ be a closed curve in $G$ which doesn't pass through $S$, which is homologous to zero in $G$. Then there are no more than finitely many points $a\in S$ for which 1;;$n_{\gamma }(a)\ne 0$, and 1;; ${\oint }_{\gamma }f(\zeta )d\zeta =2\pi i\sum _{a\in S}{n}_{\gamma }(a)\underset{z=a}{Res}f(z)$