Definition:Contour/Length/Complex Plane
< Definition:Contour | Length
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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 $C_1, \ldots, C_n$.
Sources
- 2001: Christian Berg: Kompleks funktionsteori: $\S 2.2$