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 $\gamma_i: \left[{a_i \,.\,.\, b_i}\right] \to \C$ for all $i \in \left\{ {1, \ldots, n}\right\}$.

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


The parameterization of $C$ is defined as the function $\gamma: \left[{a_1 \,.\,.\, c_n}\right] \to \C$ with:

$\gamma \restriction_{\left[{c_i \,.\,.\, c_{i + 1} }\right] } \left({t}\right) = \gamma_i \left({t}\right)$

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

Here, $\gamma \restriction_{\left[{c_i \,.\,.\, c_{i + 1} }\right] }$ denotes the restriction of $\gamma$ to $\left[{c_i \,.\,.\, c_{i + 1} }\right]$.


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


Here, $\left\{ {c_0, c_1, \ldots, c_n}\right\}$ form a subdivision of $\left[{a_1 \,.\,.\, c_n}\right]$.

The parameterization $\gamma$ is a continuous complex function that is complex-differentiable restricted to each open interval $\left({c_i \,.\,.\, c_{i + 1} }\right)$.


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