betaIntegral_symm — Mathlib

English

The Beta function evaluated at (u,1) equals 1/u when Re(u)>0.

Русский

Beta(u,1) = 1/u при Re(u)>0.

LaTeX

$$$\betaIntegral(u,1) = \dfrac{1}{u}$ for $\Re(u)>0$$$

Lean4

theorem betaIntegral_symm (u v : ℂ) : betaIntegral v u = betaIntegral u v := by
  simpa [betaIntegral, ← intervalIntegral.integral_symm, add_comm, mul_comm, sub_eq_add_neg] using
    intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
      (fun x : ℝ => (x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1)) neg_one_lt_zero.ne 1

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