\begin{definition}[Definition:Concatenation of Contours/Complex Plane]
Let $C$ and $D$ be contours in the complex plane $\C$.
Thus:
:$C$ is a (finite) sequence of directed smooth curves $\sequence {C_1, \ldots, C_n}$
:$D$ is a (finite) sequence of directed smooth curves $\sequence {D_1, \ldots, D_m}$.
Let $C_i$ be parameterized by the smooth path:
:$\gamma_i: \closedint {a_i} {b_i} \to \C$ for all $i \in \set {1, \ldots, n}$
Let $D_i$ be parameterized by the smooth path:
:$\sigma_i: \closedint {c_i} {d_i} \to \C$ for all $i \in \set {1, \ldots, m}$
Let $\map {\gamma_n} {b_n} = \map {\sigma_1} {c_1}$.
Then the '''concatenation of the contours''' $C$ and $D$, denoted $C \cup D$, is the contour defined by the (finite) sequence:
:$\sequence {C_1, \ldots, C_n, D_1, \ldots, D_m}$
\end{definition}