False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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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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FALSE: the residue theorem applies to every cycle in the ambient domain

Statement

False claim: if $f$ is meromorphic on an open set $Ω$ and $Γ$ is any cycle in $Ω ∖ S$ , then

$\int_{Γ} f (z) d z = 2 π i \sum_{a \in S} n (Γ, a) Res (f, a) .$

The missing hypothesis is that $Γ$ be null-homologous in $Ω$ .

Facts & Assumptions

Given: The punctured plane $Ω = C ∖ {0}$ , the function $f (z) = 1 / z$ , and the positively oriented unit circle $Γ$ .

[L1]

The residue theorem requires an admissible cycle, hence in particular null-homology in the ambient open set (Admissible cycles for the residue theorem, The residue theorem for a null-homologous cycle).

Refutation

technique · direct

1.1

In the punctured plane $Ω = C ∖ {0}$ , the unit circle is not null-homologous: its winding number about the omitted point $0$ is $1$ , not $0$ .

given

2.1

The function $1 / z$ is holomorphic on $Ω$ , so its pole set in the ambient domain is empty and the right-hand sum in the false claim is $0$ . However, the contour integral around the unit circle is $2 π i$ . Thus the displayed equality fails, exactly because the cycle is not null-homologous in $Ω$ .

step 1.1L1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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. V. Ahlfors, Complex Analysis, 3rd ed., Ch. 4 §5.1 (standard reference, not scraped)