False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16

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FALSE: the modulus of a contour integral always equals the absolute line integral

Statement

False claim. For every continuous $f$ and rectifiable contour $γ$ , $∣ \int_{γ} f (z) d z ∣ = \int_{γ} ∣ f (z) ∣ ∣ d z ∣ .$

Facts & Assumptions

Given: The constant function $1$ on a positively oriented circle $γ$ of radius $r > 0$ .

[L1]

A constant contour integral is the constant times the endpoint displacement (The contour integral of a constant c is c times the endpoint displacement).

[L2]

The absolute integral of $1$ is the contour length (The absolute line integral of the constant function 1 is the length of the path).

[L3]

The correct general relation is the fundamental inequality $∣ \int f d z ∣ \leq \int ∣ f ∣ ∣ d z ∣$ (The fundamental inequality: the modulus of the integral is at most the absolute line integral for rectifiable contours).

Refutation

technique · direct

1.1

Since the circle is closed, [L1] gives $\int_{γ} 1 d z = 0$ .

1.2

By [L2], its absolute integral is its positive length $2 π r$ .

L2algebra

2.1

Thus equality fails: $0 < 2 π r$ . The values still satisfy the inequality in [L3].

step 1.1step 1.2L3∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 34 results over 9 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

Lars Ahlfors, Complex Analysis, third edition, Ch. 4 §1.2 (standard reference, not scraped)