Definition:Contour/Parameterization/Complex Plane

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Definition

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

Let $C_i$ be parameterized by the smooth path $\gamma_i: \closedint {a_i} {b_i} \to \C$ 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 $\gamma: \closedint {a_1} {c_n} \to \C$ with:

$\map {\gamma \restriction_{\closedint {c_i} {c_{i + 1} } } } t = \map {\gamma_i} t$

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

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


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 $\open {c_i} {c_{i + 1} }$.


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