Gamma and Beta Functions

Real Analysis · Axiom Academy

The factorial only knows whole numbers. Two improper integrals fill in everything in between — and they run through probability, combinatorics, and physics. You know 5! = 120 . But what is (½)! ? The Gamma function answers that with an integral — and the Beta function falls right out of it. Three moves: trace it , build it , relate it . The curve that threads the factorials Define Γ by an improper integral that converges for every x > 0. Slide x and watch the marker ride the curve — at the whole numbers it lands exactly on (n−1)!, and in between it fills the gaps the factorial leaves blank. One step up the factorial ladder The factorial obeys n! = n·(n−1)!. Gamma keeps that rule for every x, not just integers — that's what makes it the right extension. Slide x and watch the two bars stay locked together. Beta: a partner that's just a ratio of Gammas The Beta function is its own improper integral on [0, 1]. Slide x and y to reshape the integrand under the curve — the shaded area IS B(x, y), and it matches Γ(x)Γ(y)/Γ(x+y) on the nose every time. One pair of integrals, three moves: trace the factorial onto a smooth curve, build it from the rule Γ(x+1) = x·Γ(x), and relate it to Beta by Γ(x)Γ(y)/Γ(x+y). That's why they show up everywhere a "count" has to make sense between the integers — the gamma and beta distributions in probability, fractional-order combinatorics, and the integrals of statistical physics.

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