Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
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. With the arc-length function of The arc-length function of a rectifiable path, 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
Used by
- The absolute line integral of the constant function 1 is the length of the path Corollary
- 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
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 52 results over 15 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- L. Ahlfors, Complex Analysis, 3rd ed., Ch. 4 (standard reference, not scraped)