How to derive the factorial of (1/2) ?

[amriths04]

i know that (1/2)! = 0.5*√pi
but i want to know how that value was got (gamma functions? if so how)?

 

[bomo321]

Re: factorial of (1/2)

i know that (1/2)! = 0.5*√pi
but i want to know how that value was got (gamma functions? if so how)?

Yes the value comes from the gamma function.

The factorial n! coincides with the gamma function at positive integer values. So if one equates the factorial function on all positive reals with the gamma function, then one can say that

(1/2)! = sqrt(pi)/2

even though the factorial function was originally only defined for the natural numbers.

Added after 14 minutes:

Noooo I'am wrong. It's Gamma(n+1) = n! and one does not equate the factorial with the gamma function. Sorry.

Added after 26 minutes:

The general definition of a factorial is

x! = Gamma(x+1)

By a little integration you can prove that

Gamma(1/2) = sqrt(pi)
Gamma(x) = (x-1) Gamma(x-1)

So the idea seems to be that

(1/2)! = Gamma(3/2) = 1/2 * Gamma(1/2)

 

[amriths04]

Re: factorial of (1/2)

yes, i am aware that Gamma(1/2) = sqrt(pi).
but i want to prove the above mathematically.

that is how ∫(t^-0.5)*(e^-t)dt between 0 and inf = sqrt(pi) ??

 

[bomo321]

Re: factorial of (1/2)

i know that (1/2)! = 0.5*√pi
but i want to know how that value was got (gamma functions? if so how)?

Damn. This is how it goes. Just calculate

Gamma(3/2)

by integrating and then just use the general definition

x! = Gamma(x+1)

for all real x.

Added after 8 minutes:

yes, i am aware that Gamma(1/2) = sqrt(pi).
but i want to prove the above mathematically.

that is how ∫(t^-0.5)*(e^-t)dt between 0 and inf = sqrt(pi) ??

I see. hmm.

Added after 25 minutes:

yes, i am aware that Gamma(1/2) = sqrt(pi).
but i want to prove the above mathematically.

that is how ∫(t^-0.5)*(e^-t)dt between 0 and inf = sqrt(pi) ??

Can't say that I see how to do the integral by just looking at it. But you can see how the integral comes
from a property of the Gamma function called the 'Euler reflection formula':

G(x)G(x-1) = pi / sin(pi*x)

by pluging in x=1/2.

 

[amir81]

factorial of (1/2)

change the variable x=z^2.
obtain the new integral and define it i.
calculate i^2 . so it will a double integral
change the variables of double integral from Cartesian to polar.
calculate i^2 . it will be pi.
now i=sqrt(pi)

 

[bomo321]

Re: factorial of (1/2)

change the variable x=z^2.
obtain the new integral and define it i.
calculate i^2 . so it will a double integral
change the variables of double integral from Cartesian to polar.
calculate i^2 . it will be pi.
now i=sqrt(pi)

Yes check out

https://en.wikipedia.org/wiki/Gaussian_integral

there you find how it's done.