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1st April 2008, 16:59 #1
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How to derive the 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)?

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1st April 2008, 18:16 #2
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Re: factorial of (1/2)
Originally Posted by amriths04
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) = (x1) Gamma(x1)
So the idea seems to be that
(1/2)! = Gamma(3/2) = 1/2 * Gamma(1/2)

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1st April 2008, 18:23 #3
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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) ??

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1st April 2008, 19:00 #4
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Re: factorial of (1/2)
Originally Posted by amriths04
Gamma(3/2)
by integrating and then just use the general definition
x! = Gamma(x+1)
for all real x.
Added after 8 minutes:
Originally Posted by amriths04
Added after 25 minutes:
Originally Posted by amriths04
from a property of the Gamma function called the 'Euler reflection formula':
G(x)G(x1) = pi / sin(pi*x)
by pluging in x=1/2.

1st April 2008, 19:44 #5
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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)
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1st April 2008, 20:40 #6
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Re: factorial of (1/2)
Originally Posted by amir81
http://en.wikipedia.org/wiki/Gaussian_integral
there you find how it's done.
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