John Ramey » Math

February 11th, 2009 by johnramey

I must have had this epiphany several times in the last few years, but I had it again this evening while being introduced to the dark side. Many students are taught that

$n! = n \times (n-1) \times \ldots 2 \times 1$

and that 0! is defined as 1. But is there some higher level mathematics in which this factorial idea is just a corollary as with most pre-graduate school mathematics. Well, there is! This is the epic Gamma function. It is defined as

$\Gamma( \alpha ) = \int_0^{\infty} y^{\alpha - 1} e^{-y} dy$ .

If $\alpha$ is a positive integer, then $\Gamma( \alpha ) = (\alpha - 1)!$ .

Here is the main event. Suppose $\alpha = 1$ ; because $\alpha$ is a positive integer, $\Gamma( \alpha ) = 0!$ But then, notice that

$\Gamma( 1 ) = \int_0^{\infty} y^{ 1 - 1 } e^{-y} dy = \int_0^{\infty} e^{-y} dy = 1$ .

Hence, our conclusion is that 0! = 1.

Posted in Math, Statistics | No Comments »