Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck 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 ∫0∞ts−1e−t dt 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
1.1

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

F1algebra
2.1

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.

step 1.1
3.1

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

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources