Can-you-please-tell-me-where-does-this-formula-come-from-And-what-means-the-factorial-of-a-non-integer-number-pi-1-2-2-4-I-ve-verified-the-above-equation-with-cal

Question Number 67631 by Hassen_Timol last updated on 29/Aug/19

Can you please tell me, where does this

formula come from ?

And what means the factorial of a non -

integer number ?

π = {(\frac{1}{2}!)}^{2} \times 4

I^{'} v e v e r i f i e d t h e a b o v e e q u a t i o n w i t h c a l c u l a t o r .

Thank you

Commented by mathmax by abdo last updated on 29/Aug/19

w e h a v e x! = Γ (x + 1) = x Γ (x) \Rightarrow (\frac{1}{2}!) = Γ (\frac{1}{2} + 1) = \frac{1}{2} Γ (\frac{1}{2}) \Rightarrow

{(\frac{1}{2}!)}^{2} \times 4 = \frac{1}{4} Γ^{2} (\frac{1}{2}) \times 4 = Γ^{2} (\frac{1}{2}) b u t

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

Commented by mathmax by abdo last updated on 29/Aug/19

Γ (\frac{1}{2}) = 2 \int_{0}^{\infty} e^{- u^{2}} d u = 2 \frac{\sqrt{π}}{2} = \sqrt{π} \Rightarrow {(\frac{1}{2}!)}^{2} \times 4 = Γ^{2} (\frac{1}{2}) = π .

Commented by Hassen_Timol last updated on 03/Sep/19

Thank you very much .

But what means the Γ (x) function and how

is it related to the x! function ?