Cauchy's residue theorem
Theorem 12.1 (Cauchy’s residue theorem). Let \(\Gamma \) be a simple closed positively oriented contour, and let \(f\) be analytic inside and on \(\Gamma \) except at finitely many points \(z_1, \dots , z_n\) inside \(\Gamma \). Then
\[ \oint _\Gamma f(z) \dd z = 2\pi i \sum _{k=1}^{n} \Res (f; z_k). \]
Source: [1, ch. 6]
Proof. Surround each \(z_k\) by a small positively oriented circle \(C_k\), with the discs disjoint and inside \(\Gamma \). By Theorem 1.3, \(\oint _\Gamma f = \sum _k \oint _{C_k} f\). On \(C_k\), \(f\) equals its Laurent series about \(z_k\), which converges uniformly there (Theorem 9.1), so it may be integrated term by term. Since
\[ \oint _{C_k} (z - z_k)^n \dd z = \begin {cases} 2\pi i, & n = -1, \\ 0, & n \neq -1, \end {cases} \]
only the \(a_{-1}\) term survives, and \(\oint _{C_k} f = 2\pi i \Res (f; z_k)\).