Hi! As many people are aware, “!” signifies a factorial in math, like 5! is 5 * 4 * 3 * 2 * 1. In my mathematics class, passing mention was made to factorials of things other than the natural numbers. Can one take the factorial of negative numbers? Decimals? Irrational numbers? Complex numbers? If one can, how is it done?
Thanks!
The Gamma Function extends the factorial function for real complex numbers.
The most popular extension of factorials is based on the Gamma function. (Note the “1 off” value of x.)
For negative integers, factorial is undefined. The function approaches +/- infinity at those points.
real and complex, obviously.
The important thing about Gamma functions is, for positive x:
Gamma(x + 1) = x · Gamma(x)
This is a lot like the factorial’s property (but again, note that it’s 1 off):
(n + 1)! = (n+1) · n!
Isn’t that handy? It means, among other things, that you only need to make a table of the values for Gamma(x) between 1 and 2. For instance, Gamma(1.6) = 0.8935. So:
Gamma(3.6) = 2.6 · 1.6 · Gamma(1.6) = 3.7170
More than you really want to know about Gamma functions
Calculate real Gammas
My first link is redundant with Desmostylus’s. Sorry!