Definition:Contour Integral/Complex

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Let $C$ be a contour defined by a finite sequence $C_1, \ldots, C_n$ of 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 $f: \Img C \to \C$ be a continuous complex function, where $\Img C$ denotes the image of $C$.

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

$\ds \int_C \map f z \rd z = \sum_{k \mathop = 1}^n \int_{a_k}^{b_k} \map f {\map {\gamma_k} t} \map {\gamma_k'} t \rd t$

Contour Integral along Closed Contour

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

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

The definition remains the same:

$\ds \oint_C \map f z \rd z := \sum_{k \mathop = 1}^n \int_{a_k}^{b_k} \map f {\map {\gamma_k} t} \map {\gamma_k'} t \rd t$

Also known as

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

Also see

  • Results about complex contour integrals can be found here.