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The real Beta--Gamma identity

Statement

For $p, q > 0$ , $B (p, q) = Γ (p) Γ (q) / Γ (p + q)$ .

Facts & Assumptions

Given: Positive reals $p, q$ and the nonnegative function $x^{p - 1} y^{q - 1} e^{- (x + y)}$ on the open first quadrant.

[F1]

If $f : D \to [0, \infty)$ is locally Riemann integrable and $(K_{j})$ is a compact Jordan exhaustion, then $\int_{D} f = \sup_{j} \int_{K_{j}} f$ , independently of the exhaustion (Every Jordan exhaustion computes a nonnegative improper multiple integral).

[F2]

If $g : U \to R^{n}$ is injective and $C^{1}$ with invertible derivative on an open set, $K \subseteq U$ is compact Jordan, and $f$ is bounded on $g (K)$ , then $f$ is integrable exactly when $(f \circ g) ∣ \det D g ∣$ is, and their integrals are equal (Change of variables for an injective $C^{1}$ map on a compact Jordan set).

[F3]

If $A, B$ are nondegenerate closed rectangles, $f$ is integrable on $A \times B$ , and every section in one coordinate order is integrable, then the ordinary iterated integral in that order exists and equals the multiple integral (Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections).

Proof

technique · direct

1.1

On compact rectangles inside the first quadrant, [F3] factors the integral as a product of one-variable integrals. Passing through rectangular exhaustion by [F1] gives total improper integral $Γ (p) Γ (q)$ .

F1F3

1.2

On compact rectangles inside $(0, \infty) \times (0, 1)$ use $x = r t$ , $y = r (1 - t)$ . The map is injective, its absolute Jacobian is $r$ , and [F2] transforms the integrand times Jacobian into $r^{p + q - 1} e^{- r} t^{p - 1} (1 - t)^{q - 1}$ .

F2algebra

2.1

A fixed nested family of such compact rectangles maps to a cofinal exhaustion of the first quadrant. By [F1] the limit is independent of this exhaustion, and by [F3] the transformed integral factors as $Γ (p + q) B (p, q)$ .

step 1.1step 1.2F1F3

3.1

Gamma is positive on the positive axis, so division of the equality in step 2.1 by $Γ (p + q)$ gives the claimed identity.

step 2.1algebra∎

Depends on

Used by

The closed form for the volume of the unit n-ball Corollary
The elementary recurrences for the real Beta function Corollary
Special values of the real Gamma and Beta functions Example

Dependency tree · two levels

49 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, §2.2(b) (standard reference, not scraped)
K. Chandrasekharan, Lectures on the Riemann Zeta-Function, Lecture 6 §1(vi) (standard reference, not scraped)