Please-formular-for-8-3-

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) = 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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