B-7-3-2-3-B-betha-function-

Question Number 126803 by john_santu last updated on 24/Dec/20

B (\frac{7}{3}, \frac{2}{3}) = ?

B = b e t h a f u n c t i o n

Answered by Dwaipayan Shikari last updated on 24/Dec/20

B (\frac{7}{3}, \frac{2}{3}) = \frac{Γ (\frac{7}{3}) Γ (\frac{2}{3})}{Γ (3)} = \frac{Γ (\frac{1}{3}) Γ (\frac{2}{3}) . \frac{4}{3} . \frac{1}{3}}{2!} = \frac{2}{9} . \frac{π}{s i n \frac{π}{3}} = \frac{4 π}{9 \sqrt{3}}

Answered by liberty last updated on 24/Dec/20

B (\frac{7}{3}, \frac{2}{3}) = \frac{Γ (\frac{7}{3}) . Γ (\frac{2}{3})}{Γ (\frac{7}{3} + \frac{2}{3})} = \frac{Γ (\frac{7}{3}) . Γ (\frac{2}{3})}{Γ (3)}

[Γ (3) = (3 - 1)! = 2]

[Γ (x + 1) = x Γ (x)]

\Leftrightarrow = \frac{\frac{4}{3} . \frac{1}{3} . Γ (\frac{1}{3}) . Γ (\frac{2}{3})}{2} = \frac{2}{9} . Γ (\frac{1}{3}) . Γ (\frac{2}{3})

[Γ (x) . Γ (1 - x) = \frac{π}{\sin (π x)}; x \notin Z]

\Leftrightarrow = \frac{2}{9} . \frac{π}{\sin (\frac{π}{3})} = \frac{4 π}{9 \sqrt{3}} = \frac{4 \sqrt{3} π}{27}

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