Definition:Contour/Parameterization

Definition
Let $\R^n$ be a real cartesian space of $n$ dimensions. Let $C_1, \ldots, C_n$ be directed smooth curves in $\R^n$.

Let $C_i$ be parameterized by the smooth path $\rho_i: \closedint {a_i} {b_i} \to \R^n$ for all $i \in \set {1, \ldots, n}$.

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

The parameterization of $C$ is defined as the function $\rho: \closedint {a_1} {c_n} \to \R^n$ with:


 * $\map {\rho \restriction_{\closedint {c_i} {c_{i + 1} } } } t = \map {\rho_i} t$

where:
 * $\ds c_i = a_1 + \sum_{j \mathop = 1}^i b_j - \sum_{j \mathop = 1}^i a_j$ for $i \in \set {0, \ldots, n}$


 * $\rho \restriction_{\closedint {c_i} {c_{i + 1} } }$ denotes the restriction of $\rho$ to $\closedint {c_i} {c_{i + 1} }$.

Note that this definition depends on the choice of parameterizations of $C_1, \ldots, C_n$.

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