DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Log-Gamma branch on the slit plane

Definition

Let $Ω = C ∖ (- \infty, 0]$ . For every $z \in Ω$ , an admissible integer $m \geq 0$ with $Re (z + m) > 0$ exists by the Archimedean property Dedekind completeness implies the Archimedean property. Choose any such $m$ and set

$ℓ (z) = ℓ_{+} (z + m) - \sum_{j = 0}^{m - 1} Log (z + j),$

where the sum is empty if $m = 0$ and $ℓ_{+}$ and $Log$ are defined in Log-Gamma branch on the right half-plane. The next lemma proves that this value is independent of the admissible $m$ and satisfies $e^{ℓ (z)} = Γ (z)$ . This is a continued branch of log-Gamma, not a separate principal $Log (Γ (z))$ .

Depends on

Used by

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

DLMF, §5.8, analytic continuation and Weierstrass product (standard reference, not scraped)
DLMF, §§5.8 and 5.11 (standard reference, not scraped)
N. M. Temme, “Numerical aspects of special functions,” Acta Numerica 17 (2008), §§2 and 3 (standard reference, not scraped)