Category:Examples of Use of Cauchy's Residue Theorem

This category contains examples of use of Cauchy's Residue Theorem.

Let $U$ be a simply connected open subset of the complex plane $\C$.

Let $a_1, a_2, \dots, a_n$ be finitely many points of $U$.

Let $f: U \to \C$ be analytic in $U \setminus \set {a_1, a_2, \dots, a_n}$.

Let $L$ be a contour in $\C$ oriented anticlockwise.

Let $\partial U_k$ denote the closed contour bounding $U_k$.

Then:

$\ds \oint_L \map f z \rd z = 2 \pi i \sum_{k \mathop = 1}^n \Res f {a_k}$

where $\Res f {a_k}$ denotes the residue at $a_k$ of $f$.