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contour integral

(Definition)

Let be a complex-valued function defined on the image of a curve $\alpha$ : $[ a, b ] \rightarrow \mathbb{C}$ , let $P = \{ a_{0}, ..., a_{n} \}$ be a partition of . We will restrict our attention to contours, i.e. curves for which the parametric equations consist of a finite number of continuously differentiable arcs. If the sum

$\displaystyle \sum_{i = 1}^{n} f(z_{i}) (\alpha(a_{i}) - \alpha(a_{i - 1})),$

where $z_{i}$ is some point $\alpha(t_{i})$ such that $a_{i - 1} \leqslant t_{i} \leqslant a_{i}$ , converges as tends to infinity and the greatest of the numbers $a_{i} - a_{i - 1}$ tends to zero, then we define the contour integral of along $\alpha$ to be the integral

$\displaystyle \int_{\alpha} f(z) dz:=\int_a^b f(\alpha(t))d\alpha(t)$

Notes

(i) If $\operatorname{Im}(\alpha)$ is a segment of the real axis, then this definition reduces to that of the Riemann integral of

between $\alpha (a)$ and $\alpha (b)$ .

(ii) An alternative definition, making use of the Riemann-Stieltjes integral, is based on the fact that the definition of this can be extended without any other changes in the wording to cover the cases where and $\alpha$ are complex-valued functions.

Now let $\alpha$ be any curve $[a, b] \rightarrow \mathbb{R}^{2}$ . Then $\alpha$ can be expressed in terms of the components $(\alpha_{1}, \alpha_{2})$ and can be associated with the complex-valued function

$\displaystyle z(t) = \alpha_{1}(t) + i \alpha_{2}(t).$

Given any complex-valued function of a complex variable, say, defined on $\operatorname{Im}(\alpha)$ we define the contour integral of along $\alpha$ , denoted by

$\displaystyle \int_{\alpha} f(z) dz$

$\displaystyle \int_{\alpha} f(z) dz = \int_{a}^{b} f(z(t)) dz(t)$

whenever the complex Riemann-Stieltjes integral on the right exists.

(iii) Reversing the direction of the curve changes the sign of the integral.

(iv) The contour integral always exists if $\alpha$ is rectifiable and is continuous.

(v) If $\alpha$ is piecewise smooth and the contour integral of along $\alpha$ exists, then

$\displaystyle \int_{\alpha} f dz = \int_{a}^{b} f(z(t)) z'(t) dt.$

"contour integral" is owned by Mathprof. [ full author list (4) | owner history (3) ]

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Other names:

complex integral

Also defines:

contour

Attachments:: estimating theorem of contour integral (Theorem) by pahio

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Cross-references: piecewise smooth, continuous, rectifiable, right, variable, complex, components, terms, Riemann-Stieltjes integral, Riemann integral, real axis, segment, integral, infinity, converges, point, sum, arcs, continuously differentiable, number, finite, equations, curves, image, function
There are 22 references to this entry.

This is version 19 of contour integral, born on 2002-07-24, modified 2007-07-24.
Object id is 3198, canonical name is ContourIntegral.
Accessed 11547 times total.

Classification:

AMS MSC:	30A99 (Functions of a complex variable :: General properties :: Miscellaneous)
	30E20 (Functions of a complex variable :: Miscellaneous topics of analysis in the complex domain :: Integration, integrals of Cauchy type, integral representations of analytic functions)

Pending Errata and Addenda

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