ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-08-29

How statement and proof provenance work

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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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The principal square root fails to respect products at $(- 1, - 1)$

Example

For the principal square root defined from the principal logarithm,

$\sqrt[Log]{z w} \neq \sqrt[Log]{z} \sqrt[Log]{w}$

at $z = w = - 1$ .

Indeed,

$\sqrt[Log]{1} = 1, \sqrt[Log]{- 1} = e^{Log (- 1) / 2} = e^{i π / 2} = i,$

$\sqrt[Log]{((- 1) (- 1))} = 1 but \sqrt[Log]{- 1} \sqrt[Log]{- 1} = i^{2} = - 1.$

Facts & Assumptions

Given: The principal-logarithm and principal-power conventions of Dictionary for holomorphic logarithm branches, the principal logarithm, and principal powers.

[F1]

The principal logarithm has $Log (- 1) = i π$ , and branch power laws can fail across the cut (Dictionary for holomorphic logarithm branches, the principal logarithm, and principal powers).

Verification

1.1

Since $1 = 1 (\cos 0 + i \sin 0)$ , the principal square root of $1$ is $1$ .

given

2.1

By [F1], $\sqrt[Log]{- 1} = e^{i π / 2} = i$ , hence $\sqrt[Log]{- 1} \sqrt[Log]{- 1} = i^{2} = - 1 \neq 1 = \sqrt[Log]{1} = \sqrt[Log]{((- 1) (- 1))}$ .

F1step 1.1algebra∎

Depends on

Dictionary for holomorphic logarithm branches, the principal logarithm, and principal powers

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

Lars V. Ahlfors, Complex Analysis, 3rd ed., Ch. 2 §3.4 The Logarithm (standard reference, not scraped)

The principal square root fails to respect products at (−1,−1)(−1,−1)