DefinitionDefinition: AI-adaptedProof: Not applicable

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.

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

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. Put $Γ = Φ \circ γ$ and $s_{γ} : = s_{Γ}$ , using the Euclidean arc-length function of The arc-length function $s_{γ} (t) = L (γ ∣_{[a, t]})$ of a rectifiable path. The complex coordinate-length theorem Complex path length equals coordinate length and piecewise speed integral, applied to every restriction, gives $s_{γ} (t) = L_{C} (γ ∣_{[a, t]})$ . Thus this is exactly the complex path arc-length function. With this notation, 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