Definition:Contour Integral/Complex

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Definition

Let $C$ be a contour defined by a finite sequence $C_1, \ldots, C_n$ of directed smooth curves.

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 $f: \operatorname{Im} \left({C}\right) \to \C$ be a continuous complex function, where $\operatorname{Im} \left({C}\right)$ denotes the image of $C$.


The contour integral of $f$ along $C$ is defined by:

$\displaystyle \int_C f \left({z}\right) \rd z = \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} f \left({\gamma_i \left({t}\right) }\right) \gamma_i' \left({t}\right) \rd t$

From Contour Integral is Well-Defined, it follows that the complex Riemann integral on the right side is defined and is independent of the parameterizations of $C_1, \ldots, C_n$.


Contour Integral along Closed Contour

Let $C$ be a closed contour in $\C$.

Then the symbol $\displaystyle \oint$ is used for the contour integral on $C$.


The definition remains the same:

$\displaystyle \oint_C f \left({z}\right) \rd z := \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} f \left({\gamma_i \left({t}\right) }\right) \gamma_i' \left({t}\right) \rd t$


Also known as

A contour integral is called a line integral or a curve integral in many texts.


Sources