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.)
(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).