Residue theorem

Let \(\Omega\subseteq\CC\) be open and let \(f\) be meromorphic on \(\Omega\) with finitely many poles \(z_1,\ldots,z_N\). Let \(\gamma\) be a piecewise smooth closed curve in \(\Omega\setminus\{z_1,\ldots,z_N\}\) whose winding number \(n_\gamma(a)\) is \(0\) for every \(a\in\CC\setminus\Omega\). Then \[ \frac{1}{2\pi i}\int_\gamma f(z)\dz=\sum_{j=1}^Nn_\gamma(z_j)\Res_{z=z_j}f. \] In particular, if \(n_\gamma(z_j)=1\) for every \(j\), then \[ \frac{1}{2\pi i}\int_\gamma f(z)\dz=\sum_{j=1}^N\Res_{z=z_j}f. \]