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factorials and the Gamma function
Posted:
Oct 25, 1997 8:31 PM


The factorial function can be generalized by noting that
n! = integral(0 to infty) of exp(x)*x^n dx.
One then defines (this is due to Euler) the Gamma function via
Gamma(k) = integral(0 to infty) of exp(x)*x^(k1) dx ,
so that when k is an integer n, Gamma(n) = (n1)!
You can turn this around to define the factorial function as
(k)! = Gamma(y+1) = integral(0 to infty) of exp(x)*x^k dx
This then allows one to find factorials of nonpositive integers.
The fact that Gamma(1/2) (= ( 1/2)!) equals sqrt(pi) [*not*, as was stated on the list, (1/2) of that] follows from the definition of the Gamma function:
Gamma(1/2) = integral(0 to infty) of exp(x)*[1/sqrt(x)] dx,
then making the change of variable y = sqrt(x), whence one gets the integral for a Gaussian, which equals sqrt(pi). (The integral itself can be done using the well known trick of using polar coordinates to evaluate integral(infty,+infty) of exp(x^2y^2) dxdy using polar coordinates.)
In general,
(n+(1/2))! = sqrt(pi)*(2n+1)!!/2^(n+1)
when n is a nonnegative integer, and you get the values for negative halfintegral values by using the result:
(x1)! = x!/x
This, of course, is one reason that one defines 0! = 1, and also shows that (n)! is undefined.
The Gamma function, and the related Beta function, come up often in statistical mechanics and in quantum mechanics, and in elementary particle physics (quarks, gluons, etc.). I think there is something called the Beta distribution in statistics which uses the Beta function, but it's been a long time, so I am not sure.
You can find more details about this in any book on complex variables, or on mathematical physics (for instance, Arfken, Mathematical Methods for Physicists).
mark snyder



