The gamma function expands the factorial function to the real (and complex) numbers
We want:
\(f(1)=1\)
\(f(x+1)=xf(x)\)
There are an infinite number of functions which fit this. The function could fluctuate between the natural numbers.
The function we use is:
\(\Gamma (z)=\int_0^\infty x^{z-1}e^{-x}dx\)