Definition:Concatenation of Contours

Definition
Let $\left[{ a \,.\,.\, b }\right]$ and $\left[{ c \,.\,.\, d }\right]$ be closed real intervals.

Let $\gamma : \left[{ a \,.\,.\, b }\right] \to \C$ and $\sigma : \left[{ c \,.\,.\, d }\right] \to \C$ be contours.

Suppose $\gamma \left({b}\right) = \sigma \left({c}\right)$.

Then the join of the contours $\gamma$ and $\sigma$, denoted $\gamma + \sigma$, is a contour $\gamma + \sigma: \left[{a \,.\,.\, b-c+d }\right]$ and is defined by:


 * $\left({\gamma + \sigma}\right) \left({t}\right) = \left\{ { \begin{array}{ll} \gamma \left({t}\right) && \forall t \in \left[{ a \,.\,.\, b }\right] \\ \sigma \left({t-b+c}\right) && \forall t \in \left[{ b \,.\,.\, b-c+d }\right] \end{array} }\right.$

It follows from Join of Contours is Contour that $\gamma + \sigma$ is a contour.

Also denoted as
Alternative notations for the join of the contours $\gamma$ and $\sigma$ are $\gamma \sigma$ and $\gamma \cup \sigma$.

None of these notations, including $\gamma + \sigma$, fully comply with standard notation.