The Residue Theorem
 Theorem 1 (The Residue Theorem): Let $A \subseteq \mathbb{C}$ be open and connected and let $f : A \to \mathbb{C}$ by analytic on $A \setminus \{ z_1, z_2, ..., z_n \}$. If $\gamma$ is a positively-oriented, simple, closed, piecewise smooth curve homotopic to a point in $A$ such that $z_1, z_2, ..., z_n \not \in \gamma$ then $\displaystyle{\int_{\gamma} f(z) \: dz = 2\pi i \sum \mathrm{The \: residues \: of \:} f \: \mathrm{in \:} \gamma}$.