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The complex line integral over a rectifiable path as a componentwise Riemann–Stieltjes integral
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, with real and imaginary parts from Real and imaginary parts, complex conjugation, and modulus. Define where the four integrals are the real Riemann–Stieltjes integrals of Riemann–Stieltjes sums, upper and lower sums, and the Riemann–Stieltjes integral. Their existence is proved in Continuous integrands have complex and absolute line integrals along every rectifiable path ↗. On a singleton parameter interval the integral is .
Depends on
Used by
- Banach algebra valued contour integral Definition
- Integration over a complex chain and the index of a chain Definition
- The winding number of a closed contour about a point off its trace Definition
- An exponential contour integral approximated by Riemann sums and evaluated by parametrization and a primitive Example
- The iterated Cauchy formula computed for z₀z₁ on a bidisc Example
- The rectifiable Riemann–Stieltjes definition on an explicit polygonal contour with corners Example
- Cauchy-kernel contour integrals may be differentiated by a direct difference-quotient estimate Lemma
- Tagged sums approximate a contour integral within oscillation times length Lemma
- Complex line integrals are linear in the integrand Proposition
- 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
- The iterated Cauchy integral formula on a polydisc Theorem
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Sources
- L. Ahlfors, Complex Analysis, 3rd ed., Ch. 4 (standard reference, not scraped)