Definition:Contour/Length/Complex Plane

Definition
Let $C_1, \ldots, C_n$ be directed smooth curves.

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

Let $C$ be the contour defined by the finite sequence $C_1, \ldots, C_n$.

The length of a contour $C$ is defined by:


 * $\displaystyle L \left({ C }\right) := \sum_{i \mathop = 1}^n \int_{a_i }^{b_i} \left\vert{ \gamma_i' \left({t}\right) }\right\vert \ \mathrm dt$

It follows from Length of Contour is Well-Defined that $L \left({ C }\right)$ is defined and independent of the parameterizations of $C_1, \ldots, C_n$.