TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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Euler's Gamma function is holomorphic on the right half-plane

Statement

Euler's Gamma function is holomorphic on the half-plane $Re z > 0$ .

Facts & Assumptions

Given: The Gamma integral on the right half-plane.

[L1]

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

[L2]

A jointly continuous finite-interval parameter integral of holomorphic functions is holomorphic (A jointly continuous finite-interval parameter integral of holomorphic functions is holomorphic).

[L3]

A locally uniform limit of holomorphic functions is holomorphic (Locally uniform limits of holomorphic functions are holomorphic and their derivatives converge locally uniformly).

Proof

technique · direct

1.1

For $n \geq 1$ , define $Γ_{n} (z) : = \int_{1 / n}^{n} t^{z - 1} e^{- t} d t .$ The integrand is jointly continuous in $(t, z)$ on $[1 / n, n] \times {z : Re z > 0}$ and holomorphic in $z$ for each fixed $t > 0$ , so [L2] makes $Γ_{n}$ holomorphic on the right half-plane.

givenL2

2.1

By [L1], on every compact subset of the right half-plane the functions $Γ_{n}$ converge uniformly to $Γ$ . Therefore [L3] makes $Γ$ holomorphic on $Re z > 0$ .

step 1.1L1L3∎

Depends on

Used by

The Beta-Gamma identity Theorem
The complex Gamma function restricts to the real Gamma function Theorem

Dependency tree · two levels

23 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)
M. Weber, Complex Analysis, §3.7 (standard reference, not scraped)