Originally Posted by

**poincare4223** Let $\displaystyle \gamma$ be a closed path in a domain $\displaystyle D$ such that $\displaystyle W(\gamma, \zeta)=0$ for all $\displaystyle \zeta \not \in D$. Suppose that $\displaystyle f(z)$ is analytic on $\displaystyle D$ except possibly at a finite number of isolated singularities $\displaystyle z_1, \ldots, z_m \in D \char`\\ \Gamma$. Show that $\displaystyle \int_{\gamma} f(z)dz = 2 \pi i \sum W(\gamma, \zeta) \text{Res}[f, z_k]$.

Hint

Consider the Laurent decomposition at each $\displaystyle z_k$.

I do not see how to get the Laurent decomposition of the $\displaystyle z_k$ to do this problem. By the way, the above notation means the winding number. It looks like perhaps the residue theorem would come into play here somehow too. I just do not see how to proceed now. Any suggestions would be very nice. Thank you.