DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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Euler's real Beta integral

Definition

For $p, q > 0$ , define $B (p, q) = \int_{0}^{1} t^{p - 1} (1 - t)^{q - 1} d t$ .

The integral is improper at both endpoints. The exact convergence theorem Euler's Beta integral converges exactly for two positive parameters ↗ proves that it exists precisely for the displayed positive parameters and therefore discharges the definition's existence obligation.

Depends on

Used by

Symmetry and the trigonometric form of the real Beta integral Proposition
Euler's Beta integral converges exactly for two positive parameters Theorem
The real Beta--Gamma identity Theorem

Dependency tree · two levels

8 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

University of Toronto MAT237Y1, The Gamma Function and the Beta Function, §2 (standard reference, not scraped)