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The absolute line integral over a rectifiable path using its arc-length function
Definition
Let be a rectifiable contour in the sense of Rectifiable complex contours, reversal, concatenation, closedness, and orientation and let be continuous on its trace. Put and , using the Euclidean arc-length function of The arc-length function of a rectifiable path. The complex coordinate-length theorem Complex path length equals coordinate length and piecewise speed integral, applied to every restriction, gives . Thus this is exactly the complex path arc-length function. With this notation, define the absolute line integral by using Riemann–Stieltjes sums, upper and lower sums, and the Riemann–Stieltjes integral. Its existence is proved in Continuous integrands have complex and absolute line integrals along every rectifiable path. On a singleton interval it is .
Depends on
- Complex path length equals coordinate length and piecewise speed integral
- Rectifiable complex contours, reversal, concatenation, closedness, and orientation
- The arc-length function $s_\gamma(t)=L(\gamma|_{[a,t]})$ of a rectifiable path
- Riemann–Stieltjes sums, upper and lower sums, and the Riemann–Stieltjes integral
Used by
- The absolute line integral of the constant function 1 is the length of the path Corollary
- Extremal length and the curve-family modulus of a path family Definition
- John domains and the John constant Definition
- Extremal length of a rectangle and of a round annulus by hand Example
- The rho-length and the extremal length are well defined Lemma
- Complex line integrals change sign under reversal and add under concatenation Proposition
- Complex and absolute line integrals are invariant under increasing continuous reparametrization Theorem
- Continuous integrands have complex and absolute line integrals along every rectifiable path Theorem
- Extremal length of the rectangle and of the round annulus Theorem
- For piecewise-C1 contours the Riemann–Stieltjes integral agrees with the parametric complex integral and the published real line integrals Theorem
- The fundamental inequality: the modulus of the integral is at most the absolute line integral for rectifiable contours Theorem
Dependency tree · two levels
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Sources
- L. Ahlfors, Complex Analysis, 3rd ed., Ch. 4 (standard reference, not scraped)