DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-30

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.

Euler's Gamma function on the right half-plane

Definition

For $z \in C$ with $Re z > 0$ , define

$Γ (z) : = \int_{0}^{\infty} t^{z - 1} e^{- t} d t,$

where for $t > 0$ one uses the real logarithm convention $t^{z - 1} : = \exp ((z - 1) \log t)$ .

The next lemma proves that the improper integral converges locally uniformly on the open right half-plane, so this definition is well posed exactly on the displayed domain.

Used by

Euler's Gamma integral converges locally uniformly on the right half-plane Lemma
The Beta-Gamma identity Theorem
The complex Gamma function restricts to the real Gamma function Theorem
The Gamma functional equation Theorem
The Weierstrass product for reciprocal Gamma Theorem

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

K. Chandrasekharan, Lectures on the Riemann Zeta-Function, Lecture 6 §1 (standard reference, not scraped)
M. Weber, Complex Analysis, §3.7 (standard reference, not scraped)