Deriving the Gamma Function

Math Calculus

Extending the Factorial Function

We know $n!$ has a restricted domain of $n\in \mathbb{N}$, but we want to extend this function to $n\in \mathbb{R}$. To do this, we define two basic properties for the gamma function:

$$\begin{gathered} n\Gamma (n)=\Gamma (n+1) \\ \Gamma (n+1)=n!,n\in \mathbb{N} \end{gathered}$$

Derivation

We know repeated differentiation can generate a factorial function, so we start by differentiating:

$${\int }_0^{\infty }{e}^{-ax}dx=\frac{1}{a}$$

Lebeniz Integral Rule allows us to differentiate inside the integral, so by repeated differentiation with respect to $a$ and cancelling out the negative sign we get:

$$\begin{gathered} {\int }_0^{\infty }x{e}^{-ax}dx=\frac{1}{a^2} \\ {\int }_0^{\infty }{x}^2{e}^{-ax}dx=\frac{2}{a^3} \\ {\int }_0^{\infty }{x}^n{e}^{-ax}dx=\frac{n!}{a^{n+1}} \end{gathered}$$

Plugging $a=1$ we get:

$$\Gamma (n)={\int }_0^{\infty }{x}^{n-1}{e}^{-x}dx$$

Plugging the definition into the above properties should affirm that this defines the gamma function.