Here's the problem: Let $G$ be the punctured unit disk (i.e missing the point $0$). Let $f:G \mapsto \mathbb{C}$ be analytic. Suppose $\gamma$ is a closed curve in $G$ homologous to $0$ (that is the winding number of $\gamma$ about any point outside of $G$ is $0$). Then what is the value of $\int_{\gamma} f$?
I said that here Cauchy's Theorem apply since $f$ is analytic and $\gamma$ is homologous to $0$ in $G$ and since $0$ is not in $G$ so the winding number of $\gamma$ about $0$ is $0$. Is this true?