TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge 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.

The complex Gamma function restricts to the real Gamma function

Statement

For every real $x > 0$ , the complex Gamma function satisfies

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

so its restriction to $(0, \infty)$ is exactly the previously defined real Gamma function.

Facts & Assumptions

Given: A real number $x > 0$ .

[L1]

The real Gamma function is defined by the same Euler integral for $x > 0$ (The real Gamma function by Euler's integral).

[L2]

That real Euler integral converges exactly for $x > 0$ (Euler's Gamma integral converges exactly for positive real parameters).

[L3]

The complex Gamma function is defined by the same integral on $Re z > 0$ (Euler's Gamma function on the right half-plane).

Proof

technique · direct

1.1

For $t > 0$ and real $x$ , the complex-analytic convention gives $t^{x - 1} = \exp ((x - 1) \log t)$ , which is the ordinary real power.

given

2.1

Since $x > 0$ , [L2] says the integral converges, and [L1] names its value as the real Gamma function. By [L3], the complex Gamma function assigns exactly the same integral to $x$ . Therefore the two constructions agree on $(0, \infty)$ .

step 1.1L1L2L3∎

Depends on

Used by

The value of Gamma at one half Corollary
The Beta-Gamma identity Theorem

Dependency tree · two levels

18 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

K. Chandrasekharan, Lectures on the Riemann Zeta-Function, Lecture 6 §1(iii) (standard reference, not scraped)