False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-28

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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 argument principle ignores multiplicity

Statement

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

Facts & Assumptions

Given: The function $f (z) = z^{2}$ and the unit circle $γ (t) = e^{i t}$ .

[L1]

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

Refutation

technique · direct

1.1given

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

2.1step 1.1L1algebra∎

Applying [L1] on the unit circle gives $\frac{1}{2 π i} \int_{γ} \frac{f^{'} (z)}{f (z)} d z = \frac{1}{2 π i} \int_{γ} \frac{2}{z} d z = 2.$ So the argument principle returns $2$ , not the number of distinct zeros. The claim is false.

Depends on

The argument principle for an admissible null-homologous cycle

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

R. W. Howell and J. H. Mathews, Complex Analysis, §8.7 (standard reference, not scraped)