Definition:Contour/Parameterization/Complex Plane

Definition

Let $C_1, \ldots, C_n$ be directed smooth curves in the complex plane $\C$.

Let $C_k$ be parameterized by the smooth path $\gamma_k: \closedint {a_k} {b_k} \to \C$ for all $k \in \set {1, \ldots, n}$.

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

The parameterization of $C$ is defined as the function $\gamma: \closedint {a_1} {c_n} \to \C$ with:

$\map {\gamma \restriction_{\closedint {c_k} {c_{k + 1} } } } t = \map {\gamma_k} t$

where $\ds c_k = a_1 + \sum_{j \mathop = 1}^k b_j - \sum_{j \mathop = 1}^k a_j$ for $k \in \set {0, \ldots, n}$.

Here, $\gamma \restriction_{\closedint {c_k} {c_{k + 1} } }$ denotes the restriction of $\gamma$ to $\closedint {c_k} {c_{k + 1} }$.

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

Here, $\set {c_0, c_1, \ldots, c_n}$ form a subdivision of $\closedint {a_1} {c_n}$.

The parameterization $\gamma$ is a continuous complex function that is complex-differentiable restricted to each open interval $\openint {c_k} {c_{k + 1} }$.