Definition:Contour/Complex Plane

Definition
Let $C_1, \ldots, C_n$ be directed smooth curves.

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

Suppose that for all $i \in \left\{ {1, \ldots, n-1}\right\}$, the endpoint of $\gamma_i$ equals the start point of $\gamma_{i+1}$.

That is, $\gamma_i \left({b_i}\right) = \gamma_{i+1} \left({a_{i + 1} }\right)$.

Then the finite sequence $C_1, \ldots, C_n$ is called a contour.

Image
The image of a contour $C$ is defined as:


 * $\displaystyle \operatorname{Im} \left({C}\right) := \bigcup_i^n \operatorname{Im} \left({\gamma_i}\right)$

where $\operatorname{Im} \left({\gamma_i}\right)$ denotes the image of $\gamma_i$.

From Reparameterization of Directed Smooth Curve Preserves Image, it follows that this definition is independent of the choice of parameterizations.

Also known as
A contour is called a directed contour, piecewise smooth path, or a piecewise smooth curve in many texts.

Some texts only use the name contour for a closed contour.

Also see

 * Directed Smooth Curve, the special case that $n = 1$.