Definition:Concatenation of Contours

Definition
Let $\R^n$ be a real cartesian space of $n$ dimensions.

Let $C$ and $D$ be contours in $\R^n$.

Thus:
 * $C$ is a (finite) sequence of directed smooth curves $\left\langle{C_1, \ldots, C_n}\right\rangle$


 * $D$ is a (finite) sequence of directed smooth curves $\left\langle{D_1, \ldots, D_m}\right\rangle$.

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

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

Let $\gamma_n \left({b_n}\right) = \sigma_1 \left({c_1}\right)$.

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


 * $\left\langle{C_1, \ldots, C_n, D_1, \ldots, D_m}\right\rangle$

Complex Plane
The definition carries over to the complex plane, in which context it is usually applied:

Also denoted as
Concatenation of contours $C$ and $D$ can also be seen denoted as:
 * $C D$
 * $C + D$
 * $C * D$

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

Also known as
Concatenation of contours can also be referred to as join of contours, but that usage is deprecated on.

Also see

 * Concatenation of Contours is Contour: demonstration that $C \cup D$ is also a contour.


 * Definition:Concatenation of Paths