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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The real Gamma function by Euler's integral

Definition

For $s > 0$ , define $Γ (s) = \int_{0}^{\infty} t^{s - 1} e^{- t} d t$ .

The integral is improper at both $0$ and $+ \infty$ . Its convergence for every $s > 0$ , and its failure for $s \leq 0$ , are proved in Euler's Gamma integral converges exactly for positive real parameters ↗, which justifies the definition on exactly the displayed domain. The integrand is positive, so $Γ (s) > 0$ throughout $(0, \infty)$ .

Depends on

Used by

Γ(1/2)=√π from the Gaussian integral Corollary
Euler's Gamma integral converges exactly for positive real parameters Theorem
The real Beta--Gamma identity Theorem
The real Gamma function is smooth and its derivatives are logarithmic moments Theorem
The real Gamma function is strictly log-convex Theorem
The real Gamma functional equation Γ(s+1)=sΓ(s) Theorem

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

University of Toronto MAT237Y1, The Gamma Function and the Beta Function, §1 (standard reference, not scraped)
K. Chandrasekharan, Lectures on the Riemann Zeta-Function, Lecture 6 §1 (standard reference, not scraped)