Definition:Contour/Image/Complex Plane

Definition
Let $C$ be a contour in $\C$ defined by the (finite) sequence $\sequence {C_1, \ldots, C_n}$ of directed smooth curves in $\C$.

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

The image of $C$ is defined as:


 * $\ds \Img C := \bigcup_{i \mathop = 1}^n \Img {\gamma_i}$

where $\Img {\gamma_i}$ denotes the image of $\gamma_i$.

If $\Img C \subseteq D$, 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 $C_1, \ldots, C_n$.