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Theorem 1 (Residue Theorem) Let \(f\) be analytic in the region \(G\) except for the isolated singularities \(a_1,a_2,\dots ,a_m\). If \(\gamma \) is a closed rectifiable curve in \(G\) which does not pass through any of the points \(a_k\) and if \(\gamma \approx 0\) in \(G\), then \[ \frac {1}{2\pi i}\int _\gamma \! f = \sum _{k=1}^m n(\gamma ;a_k)\Res (f;a_k)\,. \]