# What is the first derivative, and the integral of x!?

May 4, 2017

The factorial function is only defined for integer values, and as such it is undefined for $x \in \mathbb{R} - \mathbb{N}$ and so it does not have a derivative or integral.

Having said that the factorial function can be extended to $x \in \mathbb{R}$ by using the Gamma Function:

$\Gamma \left(z\right) = {\int}_{o}^{\infty} {x}^{z - 1} {e}^{-} x \setminus \mathrm{dx}$

Along with the relationship:

 n! = Gamma(n+1) \ \ \ AA n in NN

The graph of the Gamma function is as follows:
graph{x! [-10, 10, -5, 5]}

And the derivative of the Gamma function is obtained from the polygamma function $\psi \left(z\right)$, for which the following relationship holds:

$\psi \left(z\right) = \frac{\Gamma ' \left(z\right)}{\Gamma \left(z\right)}$

I am not sure what the integral of the Gamma function is - perhaps another member could add details.