Please formular for Γ((8/3))


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Question Number 152187 by Tawa11 last updated on 26/Aug/21

$Please formular for Γ (\frac{8}{3})$
Commented by Tawa11 last updated on 26/Aug/21

$Or generally Γ (\frac{x}{y})$
	Answered by Olaf_Thorendsen last updated on 26/Aug/21
	$• Γ(z+1) = zΓ(z) Γ((8/3)) =(5/3) Γ((5/3)) = (5/3).(2/3)Γ((2/3)) = ((10)/9)Γ((2/3)) • ∀z∈C\Z, Γ(1−z)Γ(z) = (π/(sin(πz))) • Γ(1−(1/3))Γ((1/3)) = (π/(sin((π/3)))) Γ((2/3))Γ((1/3)) = ((2π)/( (√3))) Γ((8/3)) = ((10)/9).(((2π)/( (√3)))/(Γ((1/3)))) = ((20π)/(9(√3))).(1/(Γ((1/3)))) We can prove that Γ((1/3)) = 3∫_0 ^∞ e^(−t^3 ) dt Γ((1/3)) ≈ 2,678938537 General formula : Γ(n+(1/p)) = Γ((1/p))(((pn−(p−1))!^((p)) )/p^n ) n!^((p)) denotes the pth multifactorial of n.$
	$∙ Γ (z + 1) = z Γ (z)$ $Γ (\frac{8}{3}) = \frac{5}{3} Γ (\frac{5}{3}) = \frac{5}{3} . \frac{2}{3} Γ (\frac{2}{3}) = \frac{10}{9} Γ (\frac{2}{3})$ $∙ \forall z \in C ∖ Z, Γ (1 - z) Γ (z) = \frac{π}{\sin (π z)}$ $∙ Γ (1 - \frac{1}{3}) Γ (\frac{1}{3}) = \frac{π}{\sin (\frac{π}{3})}$ $Γ (\frac{2}{3}) Γ (\frac{1}{3}) = \frac{2 π}{\sqrt{3}}$ $Γ (\frac{8}{3}) = \frac{10}{9} . \frac{\frac{2 π}{\sqrt{3}}}{Γ (\frac{1}{3})} = \frac{20 π}{9 \sqrt{3}} . \frac{1}{Γ (\frac{1}{3})}$ $We can prove that Γ (\frac{1}{3}) = 3 \int_{0}^{\infty} e^{- t^{3}} d t$ $Γ (\frac{1}{3}) \approx 2, 678938537$ $General formula :$ $Γ (n + \frac{1}{p}) = Γ (\frac{1}{p}) \frac{(p n - (p - 1))!^{(p)}}{p^{n}}$ $n!^{(p)} denotes the p th multifactorial$ $of n .$
	Commented by Tawa11 last updated on 26/Aug/21

	$Thanks sir . God bless you .$
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