Definition:Contour/Image/Complex Plane

Definition
Let $C_1, \ldots, C_n$ be directed smooth curves in the complex plane $\C$.

Let $C_i$ be parameterized by the smooth path $\gamma_i: \left[{a_i \,.\,.\, b_i}\right] \to \C$ for all $i \in \left\{ {1, \ldots, n}\right\}$. Let $C$ be the contour defined by the finite sequence $C_1, \ldots, C_n$.

The image of $C$ is defined as:


 * $\displaystyle \operatorname{Im} \left({C}\right) := \bigcup_{i \mathop = 1}^n \operatorname{Im} \left({\gamma_i}\right)$

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

If $\operatorname{Im} \left({C}\right) \subseteq D$, where $D$ is a subset of $\C$, we say that $C$ is a contour in $D$.

Also see

 * Reparameterization of Directed Smooth Curve Preserves Image, from which it follows that this definition is independent of parameterizations of $C_1, \ldots, C_n$.