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Topic: factorials and the Gamma function
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mark snyder

Posts: 1,300
Registered: 12/3/04
factorials and the Gamma function
Posted: Oct 25, 1997 8:31 PM
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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^(k-1) dx ,

so that when k is an integer n, Gamma(n) = (n-1)!

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 non-positive 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^2-y^2) dxdy using polar coordinates.)

In general,

(n+(1/2))! = sqrt(pi)*(2n+1)!!/2^(n+1)

when n is a non-negative integer, and you get the values for negative
half-integral values by using the result:

(x-1)! = 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






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