\begin{definition}[Definition:Contour/Complex Plane]
Let $C_1, \ldots, C_n$ be directed smooth curves in the complex plane $\C$.
For each $i \in \left\{ {1, \ldots, n}\right\}$, let $C_i$ be parameterized by the smooth path $\gamma_i: \left[{a_i \,.\,.\, b_i}\right] \to \C$.
For each $i \in \left\{ {1, \ldots, n - 1}\right\}$, let the endpoint of $\gamma_i$ equal the start point of $\gamma_{i + 1}$:
:$\gamma_i \left({b_i}\right) = \gamma_{i + 1} \left({a_{i + 1} }\right)$
Then the finite sequence $\left\langle{C_1, \ldots, C_n}\right\rangle$ is a '''contour'''.
If $C_1, \ldots, C_n$ are defined only by their parameterizations $\gamma_1, \ldots, \gamma_n$, then the '''contour''' can be denoted by the same symbol $\gamma$.
\end{definition}