Definition:Contour Integral/Complex

Definition
Let $C$ be a contour defined by a finite sequence $C_1, \ldots, C_n$ of directed smooth curves in the complex plane $\C$.

Let $C_k$ be parameterized by the smooth path:
 * $\gamma_k: \closedint {a_k} {b_k} \to \C$

for all $k \in \set {1, \ldots, n}$.

Let $f: \Img C \to \C$ be a continuous complex function, where $\Img C$ denotes the image of $C$.

The contour integral of $f$ along $C$ is defined by:


 * $\ds \int_C \map f z \rd z = \sum_{k \mathop = 1}^n \int_{a_k}^{b_k} \map f {\map {\gamma_k} t} \map {\gamma_k'} t \rd t$

Also see

 * Contour Integral is Well-Defined: the complex Riemann integral on the is defined and is independent of the parameterizations of $C_1, \ldots, C_n$.