contour integral

Let f be a complex-valued function defined on the image of a curve ( α: [a,b], let P={a0,,an} be a partitionPlanetmathPlanetmath ( 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


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



(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


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




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

Title contour integral
Canonical name ContourIntegral
Date of creation 2013-03-22 12:51:44
Last modified on 2013-03-22 12:51:44
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 23
Author Mathprof (13753)
Entry type Definition
Classification msc 30A99
Classification msc 30E20
Synonym complex integral
Synonym line integral
Synonym curve integral
Related topic CauchyIntegralFormula
Related topic PathIntegral
Related topic Integral
Related topic IntegralTransform
Related topic RealAndImaginaryPartsOfContourIntegral
Defines contour