Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28
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FALSE: the argument principle ignores multiplicity

Statement

False claim: the argument principle counts zeros of a holomorphic function without multiplicity.

Facts & Assumptions

Given: The function f(z)=z2 and the unit circle γ(t)=eit.

[L1]

The argument principle counts zeros with multiplicity (The argument principle for an admissible null-homologous cycle).

Refutation

technique · direct
1.1

The function f(z)=z2 has exactly one distinct zero, namely 0, but that zero has multiplicity 2.

given
2.1

Applying [L1] on the unit circle gives 12πi∫γf′(z)f(z) dz=12πi∫γ2z dz=2. So the argument principle returns 2, not the number of distinct zeros. The claim is false.

step 1.1L1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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