DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

How 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.

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The absolute line integral over a rectifiable path using its arc-length function

Definition

Let $γ : [a, b] \to C$ be a rectifiable contour in the sense of Rectifiable complex contours, reversal, concatenation, closedness, and orientation and let $f$ be continuous on its trace. With $s_{γ}$ the arc-length function of The arc-length function $s_{γ} (t) = L (γ ∣_{[a, t]})$ of a rectifiable path, define the absolute line integral by $\int_{γ} ∣ f (z) ∣ ∣ d z ∣ : = \int_{a}^{b} ∣ f (γ (t)) ∣ d s_{γ} (t),$ 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 $0$ .

Depends on

Used by

Dependency tree · two levels

19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

L. Ahlfors, Complex Analysis, 3rd ed., Ch. 4 (standard reference, not scraped)