请输入您要查询的字词:

 

单词 ContourIntegral
释义

contour integral


Let f be a complex-valued function defined on the image of a curve (http://planetmath.org/Curve) α: [a,b], let P={a0,,an} be a partitionPlanetmathPlanetmath (http://planetmath.org/Partition3) of [a,b]. 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

i=1nf(zi)(α(ai)-α(ai-1)),

where zi is some point α(ti) such that ai-1tiai, converges as n tends to infinityMathworldPlanetmath and the greatest of the numbers ai-ai-1 tends to zero, then we define the contour integral of f along α to be the integral

αf(z)𝑑z:=abf(α(t))𝑑α(t)

Notes

(i) If Im(α) is a segment of the real axis, then this definition reduces to that of the Riemann integral of f(x) between α(a) and α(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 f and α are complex-valued functions.

Now let α be any curve [a,b]2. Then α can be expressed in terms of the components (α1,α2) and can be associated with the complex-valued function

z(t)=α1(t)+iα2(t).

Given any complex-valued function of a complex variable, f say, defined on Im(α) we define the contour integral of f along α, denoted by

αf(z)𝑑z

by

αf(z)𝑑z=abf(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 α is rectifiable and f is continuousMathworldPlanetmathPlanetmath.

(v) If α is piecewise smooth and the contour integral of f along α exists, then

αf𝑑z=abf(z(t))z(t)𝑑t.
Titlecontour integral
Canonical nameContourIntegral
Date of creation2013-03-22 12:51:44
Last modified on2013-03-22 12:51:44
OwnerMathprof (13753)
Last modified byMathprof (13753)
Numerical id23
AuthorMathprof (13753)
Entry typeDefinition
Classificationmsc 30A99
Classificationmsc 30E20
Synonymcomplex integral
Synonymline integral
Synonymcurve integral
Related topicCauchyIntegralFormula
Related topicPathIntegral
Related topicIntegral
Related topicIntegralTransform
Related topicRealAndImaginaryPartsOfContourIntegral
Definescontour
随便看

 

数学辞典收录了18232条数学词条,基本涵盖了常用数学知识及数学英语单词词组的翻译及用法,是数学学习的有利工具。

 

Copyright © 2000-2023 Newdu.com.com All Rights Reserved
更新时间:2025/5/4 19:39:39