# Definition:Contour/Length

## Definition

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

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

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

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

- $\displaystyle L \left({C}\right) := \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} \left\vert{\rho_i' \left({t}\right) }\right\vert \rd t$

### 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 $\left\langle{C_1, \ldots, C_n}\right\rangle$ of directed smooth curves in $\C$.

Let $C_i$ be parameterized by the smooth path $\gamma_i: \left[{a_i\,.\,.\,b_i}\right] \to \C$ for all $i \in \left\{ {1, \ldots, n}\right\}$.

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

- $\displaystyle L \left({C}\right) := \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} \left\vert{\gamma_i' \left({t}\right) }\right\vert \rd t$

## Also see

- Length of Contour is Well-Defined: $L \left({C}\right)$ is defined and independent of the parameterizations of $C_1, \ldots, C_n$.