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May 29th, 2019, 10:23 PM   #1
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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.
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May 29th, 2019, 11:01 PM   #2
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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
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May 29th, 2019, 11:06 PM   #3
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Quote:
Originally Posted by idontknow View Post
Why gamma(1/2) is $\displaystyle 1/2 \cdot sqrt(\pi) $?
It isn't.
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May 30th, 2019, 02:46 AM   #4
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Quote:
Originally Posted by idontknow View Post
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
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Last edited by topsquark; May 30th, 2019 at 02:54 AM.
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