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CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16
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The integral of a continuous complex derivative over every closed rectifiable contour is zero

Statement

If F is holomorphic on an open set, F′ is continuous there, and γ is a closed rectifiable contour in that set, then ∫γF′(z) dz=0.

Facts & Assumptions

Given: A function F and a closed rectifiable contour γ as in the Statement.

[L1]

The contour fundamental theorem gives ∫γF′ dz=F(γ(b))−F(γ(a)) (The line integral of a continuous function admitting a primitive is that primitive's endpoint increment along every rectifiable path).

[L2]

A contour is closed exactly when its endpoint values agree (Rectifiable complex contours, reversal, concatenation, closedness, and orientation).

Proof

technique · direct
1.1L1L2algebra

Apply [L1] and then [L2]: the endpoint increment is F(γ(a))−F(γ(a))=0.

2.1step 1.1∎

Thus the integral vanishes, including for a constant closed contour.

Depends on

Dependency tree · 2 levels

Sources