Definition:Contour/Complex Plane

Definition
Let $a, b \in \R : a < b$.

Let $\gamma : \left[{ a \,.\,.\, b }\right] \to \C$ be a path.

Then, $\gamma$ is a contour iff there exist a natural number $n \in \N_{ >0 }$ and $a_0, a_1, \ldots , a_n \in \left[{ a \,.\,.\, b }\right] $ such that:


 * $(1):\quad $ $a_0, a_1, \ldots , a_n$ form a subdivision of $\left[{ a \,.\,.\, b }\right]$.
 * $(2):\quad $ For all $i \in \left\{{ 1, 2, \ldots, n }\right\}$, the path $\gamma \restriction_{ I_i }$ is a smooth path, where $I_i = \left[{ a_{ i - 1 } \,.\,.\, a_i }\right]$ is a closed interval.

Here, $\gamma \restriction_{ I_i } : I_i \to \C$ denotes the restriction of $\gamma$ to $I_i$.

Also known as
A contour is called a piecewise smooth path in many texts.

Also see

 * Smooth Path (Complex Analysis), the special case that $n = 1$.