TITLE: Defining the factorial of a real number
QUESTION [6 upvotes]: 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?

REPLY [7 votes]: 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.