Definition:Contour/Length/Complex Plane

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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 $C_i$ be parameterized by the smooth path $\gamma_i: \closedint {a_i} {b_i} \to \C$ for all $i \in \set {1, \ldots, n}$.

The length of $C$ is defined as:

$\ds \map L C := \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} \size {\map {\gamma_i'} t} \rd t$

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