Definition:Contour/Image/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 $i \in \left\{ {1, \ldots, n}\right\}$.

Let $C$ be the contour defined by the finite sequence $C_1, \ldots, C_n$.

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.