False statementConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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FALSE: Euler's real Gamma integral converges at the nonpositive integers

Statement

False claim: the Euler integral $\int_{0}^{\infty} t^{s - 1} e^{- t} d t$ converges when $s$ is a nonpositive integer and thereby defines a real Gamma value there.

Facts & Assumptions

Given: A nonpositive integer $s$ .

[F1]

The Euler integral converges if and only if $s > 0$ (Euler's Gamma integral converges exactly for positive real parameters).

Refutation

technique · direct

1.1F1algebra

Since $s \leq 0$ , the reverse direction of [F1] says that the Euler integral diverges, already at its endpoint $0$ .

2.1step 1.1

The real Gamma function defined by Euler's integral therefore has no value at this $s$ ; a different continuation would not be convergence of this integral.

3.1step 1.1step 2.1∎

As the argument applies to every nonpositive integer, the claim is false.

Depends on

Euler's Gamma integral converges exactly for positive real parameters

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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(i) (standard reference, not scraped)