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 May 29th, 2019, 10:23 PM #1 Senior Member   Joined: Dec 2015 From: somewhere Posts: 541 Thanks: 82 Explanation for factorial ! Why gamma(1/2) is $\displaystyle 1/2 \cdot sqrt(\pi)$? Does $\displaystyle (1/2)!$ make sense or applied in physics? If so , then is (1/2)! the product of all real numbers in interval (0,1]? Last edited by skipjack; May 29th, 2019 at 11:06 PM.
 May 29th, 2019, 11:01 PM #2 Senior Member   Joined: Feb 2016 From: Australia Posts: 1,826 Thanks: 646 Math Focus: Yet to find out. They show up all over the place and it's probably not as mysterious as you think (though the more you look, the more intricate it becomes I suppose). https://en.wikipedia.org/wiki/Gamma_function Thanks from topsquark and idontknow
May 29th, 2019, 11:06 PM   #3
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Quote:
 Originally Posted by idontknow Why gamma(1/2) is $\displaystyle 1/2 \cdot sqrt(\pi)$?
It isn't.

May 30th, 2019, 02:46 AM   #4
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Quote:
 Originally Posted by idontknow Why gamma(1/2) is $\displaystyle 1/2 \cdot sqrt(\pi)$? Does $\displaystyle (1/2)!$ make sense or applied in physics? If so , then is (1/2)! the product of all real numbers in interval (0,1]?
The factorial function and the gamma function are closely related. Basically:
$\displaystyle \Gamma (n) = (n - 1)!$. We have $\displaystyle \Gamma (1/2) = \sqrt{ \pi }$ and $\displaystyle (1/2)! = \dfrac{\sqrt{\pi}}{2}$.

(1/2)! can't be the product of all real numbers in (0, 1]. That would be....hard to calculate. One definition of the Gamma function where the argument does not need to be an integer is
$\displaystyle \Gamma (z) = \int _0^{\infty} e^{z - 1} x^z ~ dx$. Here z is, in general, a complex number such that the real part of z is greater than 0. Here's a link to see a graph that includes negative values of z.

The Gamma function has many many uses in both Mathematics and Physics.

-Dan

Last edited by topsquark; May 30th, 2019 at 02:54 AM.

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