Definition:Concatenation of Contours

Definition
Let $C$ and $D$ be contours.

That is, $C$ is a finite sequence of directed smooth curves $C_1, \ldots, C_n$.

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

Similarly, $D$ is a finite sequence of directed smooth curves $D_1, \ldots, D_m$.

Let $D_i$ be parameterized by the smooth path $\sigma_i: \left[{c_i\,.\,.\,d_i}\right] \to \C$ for all $i \in \left\{ {1, \ldots, m}\right\}$.

Suppose $\gamma_i \left({b_n}\right) = \sigma \left({a_1}\right)$.

Then the concatenation of the contours $C$ and $D$, denoted $C \cup D$, is the contour that is defined by the sequence:


 * $C_1, \ldots, C_n, D_1, \ldots, D_m$

It follows from Concatenation of Contours is Contour that $C \cup D$ is a contour.

Also denoted as
Alternative notations for the concatenation of the contours $C$ and $D$ are $CD$ and $C + D$.

None of these notations, including $C \cup D$, fully comply with standard notation.