beta_eq_betaIntegralReal — Mathlib

English

The beta function equals the real part of the betaIntegral evaluated at the real arguments.

Русский

Бета-функция равна вещественной части интеграла бета, вычисленного на вещественных аргументах.

LaTeX

$$$\beta(\alpha,\beta) = \operatorname{Re}(\betaIntegral(\alpha,\beta)).$$$

Lean4

/-- Relation between the beta function and the gamma function over the reals. -/
theorem beta_eq_betaIntegralReal (α β : ℝ) (hα : 0 < α) (hβ : 0 < β) : beta α β = (betaIntegral α β).re :=
  by
  rw [betaIntegral_eq_Gamma_mul_div]
  · simp_rw [beta, ← ofReal_add α β, Gamma_ofReal]
    norm_cast
  all_goals simpa

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