# Definition:Contour/Length/Complex Plane

## Definition

Let $C$ be a contour in $\C$ defined by the (finite) sequence $\sequence {C_1, \ldots, C_n}$ of directed smooth curves in $\C$.

Let $C_k$ be parameterized by the smooth path $\gamma_k: \closedint {a_k} {b_k} \to \C$ for all $k \in \set {1, \ldots, n}$.

The length of $C$ is defined as:

$\ds \map L C := \sum_{k \mathop = 1}^n \int_{a_k}^{b_k} \size {\map {\gamma_k'} t} \rd t$

It follows from Length of Contour is Well-Defined that $\map L C$ is defined and independent of the parameterizations of $C_1, \ldots, C_n$.