\begin{definition}[Definition:Contour/Image/Complex Plane]
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 $D$'''.
\end{definition}