# Definition:Contour/Closed/Complex Plane

< Definition:Contour | Closed(Redirected from Definition:Closed Contour (Complex Plane))

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## Definition

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 $i \in \left\{ {1, \ldots, n}\right\}$.

$C$ is a **closed contour** if and only if the start point of $C$ is equal to the end point of $C$:

- $\gamma_1 \left({a_1}\right) = \gamma_n \left({b_n}\right)$

## Also see

- Reparameterization of Directed Smooth Curve Maps Endpoints To Endpoints, from which it follows that this definition is independent of the parameterizations of $C_1$ and $C_n$.

## Also known as

A **closed contour** is called a **loop** in some texts.

Some texts define a **contour** to be what $\mathsf{Pr} \infty \mathsf{fWiki}$ refers to as a **closed contour**.

## Sources

- 2001: Christian Berg:
*Kompleks funktionsteori*: $\S 2.2$