The gamma and beta functions

Expanding functions of natural numbers

Gamma function

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:

\(\Gamma (z)=\int_0^\infty x^{z-1}e^{-x}dx\)

Beta function

The beta function expand the binomial coefficient formula to the real (and complex) numbers.

We want to expand the binomial coefficient function.

\((\dfrac{n}{k})=\dfrac{n!}{k!(n-k)!}\)

We do this as:

\(B(x, y)=\dfrac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}\)