Question

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

Answer 1

The factorial function is only defined for integer values, and as such it is undefined for `x in RR - NN` and so it does not have a derivative or integral.

Having said that the factorial function can be extended to `x in RR` by using the Gamma Function:

`Gamma(z) = int_o^oo x^(z-1)e^-x \ 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(z)`, for which the following relationship holds:

`psi(z) = (Gamma'(z))/(Gamma(z))`

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