TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-30

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

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The Gamma functional equation

Statement

For every $z$ with $Re z > 0$ ,

$Γ (z + 1) = z Γ (z) .$

Facts & Assumptions

Given: A complex number $z$ with $Re z > 0$ .

[L1]

Integration by parts holds on compact intervals (If $u, v$ are differentiable on $[a, b]$ with $u^{'}, v^{'}$ integrable, then $\int_{a}^{b} u v^{'} = u (b) v (b) - u (a) v (a) - \int_{a}^{b} u^{'} v$ ).

[L2]

The Gamma integral converges locally uniformly on right-half-plane compact sets (Euler's Gamma integral converges locally uniformly on the right half-plane).

[L3]

Euler's Gamma function is the improper integral $\int_{0}^{\infty} t^{z - 1} e^{- t} d t$ on $Re z > 0$ (Euler's Gamma function on the right half-plane).

Proof

technique · direct

1.1

For $0 < ε < R$ , apply [L1] on $[ε, R]$ with $u (t) = t^{z}$ and $v^{'} (t) = e^{- t}$ . This gives $\int_{ε}^{R} t^{z} e^{- t} d t = [- t^{z} e^{- t}]_{ε}^{R} + z \int_{ε}^{R} t^{z - 1} e^{- t} d t .$

givenL1

2.1

Since $Re z > 0$ , one has $∣ ε^{z} e^{- ε} ∣ = ε^{Re z} e^{- ε} \to 0$ as $ε ↓ 0$ , and $∣ R^{z} e^{- R} ∣ = R^{Re z} e^{- R} \to 0$ as $R ↑ \infty$ . Passing to the improper limits in step 1.1 and using [L2] and [L3] gives $Γ (z + 1) = z Γ (z)$ .

step 1.1L2L3algebra∎

Depends on

Used by

Gamma at the positive integers Corollary
Half-integer and negative-half-integer values of Gamma Example
FALSE: the Gamma recurrence and factorial values characterize Gamma False statement
Meromorphic continuation of Gamma Theorem

Dependency tree · two levels

15 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(iv) (standard reference, not scraped)
M. Weber, Complex Analysis, §3.7 (standard reference, not scraped)