CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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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.
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The absolute line integral of the constant function 1 is the length of the path

Statement

For every rectifiable contour $γ$ , $\int_{γ} 1 ∣ d z ∣ = L (γ) .$

Facts & Assumptions

Given: A rectifiable contour $γ : [a, b] \to C$ .

[L1]

The absolute line integral is $\int_{a}^{b} ∣ f (γ (t)) ∣ d s_{γ} (t)$ (The absolute line integral over a rectifiable path using its arc-length function).

[L2]

The arc-length function satisfies $s_{γ} (a) = 0$ and $s_{γ} (b) = L (γ)$ (The arc-length function $s_{γ} (t) = L (γ ∣_{[a, t]})$ of a rectifiable path).

Proof

technique · direct

1.1

With $f \equiv 1$ , [L1] becomes $\int_{a}^{b} 1 d s_{γ} = s_{γ} (b) - s_{γ} (a)$ .

L1algebra

2.1

Substitute [L2] to obtain $L (γ)$ . Singleton and constant paths have both sides $0$ .

step 1.1L2∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 25 results over 10 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)