Gammaℝ_add_two — Mathlib

English

For s > 0, log(doublingGamma(s)) equals log Γ(s/2) + log Γ(s/2 + 1/2) + s log 2 − log(2√π).

Русский

Для s > 0 логарифм doublingGamma(s) равен сумме log Γ(s/2) + log Γ(s/2 + 1/2) плюс s log 2 минус log(2√π).

LaTeX

$$$ \log(\mathrm{doublingGamma}(s)) = \log\Gamma\left(\dfrac{s}{2}\right) + \log\Gamma\left(\dfrac{s}{2}+\tfrac{1}{2}\right) + s \log 2 - \log\bigl(2 \sqrt{\pi}\bigr) $$$

Lean4

theorem Gammaℝ_add_two {s : ℂ} (hs : s ≠ 0) : Gammaℝ (s + 2) = Gammaℝ s * s / 2 / π :=
  by
  rw [Gammaℝ_def, Gammaℝ_def, neg_div, add_div, neg_add, div_self two_ne_zero,
    Gamma_add_one _ (div_ne_zero hs two_ne_zero), cpow_add _ _ (ofReal_ne_zero.mpr pi_ne_zero), cpow_neg_one]
  field_simp

Похожие утверждения