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
[latex]n! = n \times (n-1) \times \ldots 2 \times 1[/latex]
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
[latex]\Gamma( \alpha ) = \int_0^{\infty} y^{\alpha – 1} e^{-y} dy[/latex].
If [latex]\alpha[/latex] is a positive integer, then [latex]\Gamma( \alpha ) = (\alpha – 1)![/latex].
Here is the main event. Suppose [latex]\alpha = 1[/latex]; because [latex]\alpha[/latex] is a positive integer, [latex]\Gamma( \alpha ) = 0![/latex] But then, notice that
[latex]\Gamma( 1 ) = \int_0^{\infty} y^{ 1 – 1 } e^{-y} dy = \int_0^{\infty} e^{-y} dy = 1[/latex].
Hence, our conclusion is that 0! = 1.