Lecture 18: The Beta-Gamma Identity

Proposition (iii)

For $x > 0$ and $y > 0$, $$\int_0^1 t^{x-1}(1-t)^{y-1}\,dt = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}.$$

This extends to complex parameters: for $\operatorname{Re} z > 0$ and $\operatorname{Re} w > 0$, $$\int_0^1 t^{z-1}(1-t)^{w-1}\,dt = \frac{\Gamma(z)\Gamma(w)}{\Gamma(z+w)}.$$

Implementation notes

The complex Beta integral B(z, w) = ∫ t in 0..1, t^(z-1) (1-t)^(w-1) dt is defined in Mathlib as Complex.betaIntegral z w. The identity B(z,w) = Γ(z) Γ(w) / Γ(z+w) is Complex.betaIntegral_eq_Gamma_mul_div.

For the real version, we express the integral ∫ t in 0..1, t^(x-1) (1-t)^(y-1) dt using real-valued rpow and reduce to the complex result via Complex.ofReal_cpow and intervalIntegral.integral_ofReal.

theorem Real.betaIntegral_eq_Gamma_mul_div (x y : ℝ) (hx : 0 < x) (hy : 0 < y) : ∫ (t : ℝ) in 0..1, t ^ (x - 1) * (1 - t) ^ (y - 1) = Gamma x * Gamma y / Gamma (x + y)

Proposition (iii), Lecture 18 (real version). For x > 0 and y > 0, the Beta integral ∫ t in 0..1, t ^ (x-1) * (1-t) ^ (y-1) equals Γ(x) Γ(y) / Γ(x+y).

theorem Complex.betaIntegral_eq_Gamma_mul_div_Lecture18 (z w : ℂ) (hz : 0 < z.re) (hw : 0 < w.re) : z.betaIntegral w = Gamma z * Gamma w / Gamma (z + w)

Proposition (iii), Lecture 18 (complex extension). For Re z > 0 and Re w > 0, the Beta integral B(z, w) = ∫ t in 0..1, t ^ (z-1) * (1-t) ^ (w-1) equals Γ(z) Γ(w) / Γ(z+w). Here Complex.betaIntegral z w denotes this integral.