I'm brushing up on my complex for an upcoming qual, and one of the questions had me use an alternate form of the residue theorem:
$$\oint_{\partial \Omega} \frac{f(\zeta)}{z-\zeta} d \zeta = 2 \pi i \sum_{j=1}^{n}\text{Res}(f,z_j)$$
for $f$ meromorphic on a bounded, star-shaped domain $D$ with boundary $\Gamma$, and $z \in \Omega$. I'm pretty sure it involves that the index of any path is one, but I can't figure out how to put it together.