Definition:Concatenation of Contours

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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 $\sequence {C_1, \ldots, C_n}$
$D$ is a (finite) sequence of directed smooth curves $\sequence {D_1, \ldots, D_m}$.

Let $C_i$ be parameterized by the smooth path:

$\gamma_i: \closedint {a_i} {b_i} \to \R^n$ for all $i \in \set {1, \ldots, n}$

Let $D_i$ be parameterized by the smooth path:

$\sigma_i: \closedint {c_i} {d_i} \to \R^n$ 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:

$\sequence {C_1, \ldots, C_n, D_1, \ldots, D_m}$


Complex Plane

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


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:

$\sequence {C_1, \ldots, C_n, D_1, \ldots, D_m}$


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 $\mathsf{Pr} \infty \mathsf{fWiki}$.


Also see


Linguistic Note

The word concatenation derives from the Latin word catena for chain.

However, the end result of such an operation is not to be confused with a (set theoretical) chain.