Alphabeta MathDefinitionDefinition: 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 with , define
where for one uses the real logarithm convention .
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
- The completed zeta function Λ(s)=π^-s/2Γ(s/2)ζ(s) Definition
- Euler's Gamma integral converges locally uniformly on the right half-plane Lemma
- K-type eigenvalues of A(nu): recurrence, closed form and nonvanishing Lemma
- The Beta-Gamma identity Theorem
- The completed zeta function has its Mellin-theta integral representation on Res>1 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)