Definition:Contour/Length/Complex Plane

Definition
The length of a contour $\gamma : \left[{a \,.\,.\, b}\right] \to \C$ is defined by:


 * $\displaystyle L \left({ \gamma }\right) := \sum_{i \mathop = 1}^n \int_{a_{i-1} }^{a_i} \left\vert{ \gamma' \restriction_{I_i} \left({t}\right) }\right\vert \ \mathrm dt$

It follows from Length of Contour is Well-Defined that $L \left({ \gamma }\right)$ is defined and independent of the subdivision $a_0, a_1, \ldots, a_n$ of $\left[{a \,.\,.\, b}\right]$. Here, $I_i$ denotes the closed real interval $\left[{a_{i-1} \,.\,.\, a_i}\right]$.