# Definition:Contour/Length

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## Definition

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Let $\R^n$ be a real cartesian space of $n$ dimensions.

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

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

The **length** of $C$ is defined as:

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

### Complex Plane

The definition carries over to the complex plane, in which context it is usually applied:

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:

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

## Also see

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