I'm curious, how is the factorial of a real number defined? Intuitively, it should be:
$x! = 0$ if $x \leq 1$
$x! = \infty$ if $x >1$
Since it would be the product of all real numbers preceding it, however, when I plug $\pi!$ into my calculator, I get an actual value: $7.18808272898$
How is that value determined?
The Gamma Function defined by \begin{equation*} \Gamma(z) = \int_{0}^{\infty}t^{z-1}e^{-t}dt \end{equation*} satisfies $\Gamma (n) = (n-1)!$ for all $n \in \mathbb{N}$, $n \geq 1$, so its an extension of the factorial to real and complex numbers.
The Gamma Function is the generalization. It is a function with a real argument with the property that $\Gamma(x+1)=x!$ whenever $x$ is a natural or zero. It can also take complex arguments. It is not defined on the non-positive integers, where it has poles.
The Gamma Function, $\Gamma(x)$ can be represented by the integral
$$\Gamma(x)=\int_0^\infty t^{x-1}e^{-t}\,dt$$
for $x>0$.
Note that if $x=n$, where $n$ is an integer, then it is easy to show by integrating by parts that
$$\Gamma (n+1)=n!$$
