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 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
- Contour Integral is Well-Defined: the complex Riemann integral on the right hand side is defined and is independent of the parameterizations of $C_1, \ldots, C_n$.
- Results about complex contour integrals can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): contour integral
- 2001: Christian Berg: Kompleks funktionsteori $\S 2.2$
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): contour integral