A Gamma Function

We want to generalize the factorial function which was \[n! = n(n-1)(n-2)\dots(2)(1)\] for \(n \in \{0, 1, 2, \dots\}\) and we define \(0!=1\) for notational convenience because that is what we need in many formulas. Notice that \(1! = 1\), \(2!=2\) and \(3! = 3 * 2!\).

The problem is that this is only defined for the whole numbers (natural numbers and zero). We want to define a function that allows us to expand this function to the positive real numbers.

Consider the funtion: \[\Gamma(x) = \int_0^\infty z^{x-1} e^{-z} \,dz\]

First we show that \(\Gamma(x) = (x-1)\Gamma(x-1)\) utilizing integration by parts.

\[\begin{aligned} u = z^{x-1} &\;\;\; & dv = e^{-z}\,dz \\ du = (x-1)z^{x-2}\,dx &\;\;\; & v = -e^{-z} \end{aligned}\]

and therefore

\[\begin{aligned} \Gamma(x) &= \int_0^\infty z^{x-1} e^{-z} \,dz \\ &= \int_0^\infty u \;dv \\ &= uv\vert_0^\infty - \int_0^\infty v\, du \\ &= -z^{x-1}e^{-z}\vert_{z=0}^\infty + \int_0^\infty (x-1)z^{x-2}e^{-z} \, dz\\ &= (x-1) \int_0^\infty z^{x-2}e^{-z}\,dz\\ &= (x-1) \Gamma(x-1) \end{aligned}\]

Next we notice that \(\Gamma(1) = 1 = 0!\) via the following integration \[\Gamma(1) = \int_0^\infty z^{1-1}e^{-z}\,dz = \int_0^\infty e^{-z} \,dz = -e^{-z} \vert_0^\infty = 1\]

Using these bits, we can see that \[\Gamma(2) = 1\cdot \Gamma(1) = 1 = 1!\] \[\Gamma(3) = 2 \Gamma(2) = 2 = 2!\] \[\Gamma(4) = 3 \Gamma(3) = 3 \cdot 2! = 3!\] and for positive integers \(n\), \[\Gamma(n) = (n-1)!\] which can be re-arranged to see that \[n! = n \Gamma(n)\]

We conclude by noting that \[\Gamma\left( \frac{1}{2} \right) = \sqrt \pi\] (This result can be obtained by looking at the square of the integral and then doing a trig substitution.)

A graph of this on the postive real numbers is given below:

Finally we note that there is a reasonable approximation (known as Stirling’s approximation) to the function for large values of \(x\) as \[\Gamma(x+1) \approx \sqrt{ 2 \pi x } \left( \frac{x}{e} \right)^x\]