7-2-

Question Number 108239 by Dwaipayan Shikari last updated on 15/Aug/20

(\frac{7}{2})! = ?

Answered by abdomsup last updated on 15/Aug/20

Γ (x + n) = Γ (x + n - 1 + 1)

= (x + n - 1) Γ (x + n - 2 + 1)

= (x + n - 1) (x + n - 2) Γ (x + n - 2)

= (x + n - 1) (x + n - 2) (x + n - 3) Γ (x + n - 3)

\Rightarrow (\frac{7}{2})! = Γ (\frac{7}{2} + 1)

= Γ (\frac{1}{2} + 4) = (\frac{1}{2} + 3) (\frac{1}{2} + 2) (\frac{1}{2} + 1) Γ (\frac{1}{2} + 1)

= \frac{7}{2} . \frac{5}{2} . \frac{3}{2} . \frac{1}{2} Γ (\frac{1}{2})

= \frac{35.3}{8} Γ (\frac{1}{2}) = \frac{105}{8} Γ (\frac{1}{2})

Γ (x) = \int_{0}^{\infty} t^{x - 1} e^{- t} d t \Rightarrow

Γ (\frac{1}{2}) = \int_{0}^{\infty} \frac{e^{- t}}{\sqrt{t}} d t =_{t = u^{2}} \int_{0}^{\infty} \frac{e^{- u^{2}}}{u} (2 u) d u

= 2 \int_{0}^{\infty} e^{- u^{2}} d u = 2 . \frac{\sqrt{π}}{2} = \sqrt{π} \Rightarrow

(\frac{7}{2})! = \frac{105 \sqrt{π}}{8}

Commented by abdomsup last updated on 15/Aug/20

s o r r y (\frac{7}{2})! = \frac{105 \sqrt{π}}{16}

Commented by abdomsup last updated on 15/Aug/20

g e n e r a l y w e h a v e

Γ (x + n) = (x + n - 1) (x + n - 2)

\dots . (x + n - k) Γ (x + n - k)

k = n \Rightarrow Γ (x + n) = (x + n - 1) (x + n - 2)

\dots \dots (x + 1) x Γ (x)

e x n = 4 \Rightarrow

Γ (x + 4) = (x + 3) (x + 2) (x + 1) Γ (x)

x = \frac{1}{m} \Rightarrow Γ (\frac{1}{m} + 4)

= (\frac{1}{m + 3}) (\frac{1}{m} + 2) (\frac{1}{m} + 1) \frac{1}{m} Γ (\frac{1}{m})

Commented by abdomsup last updated on 15/Aug/20

e r r o r o f t y p o

Γ (\frac{1}{m} + 4) = (\frac{1}{m} + 3) (\frac{1}{m} + 2) (\frac{1}{m} + 1) \frac{1}{m} Γ (\frac{1}{m})

Commented by mathmax by abdo last updated on 15/Aug/20

Γ (x + 4) = (x + 3) (x + 2) (x + 1) x Γ (x)