I was messing around with my TI-84 Plus Silver Edition calculator and discovered that it will actually give me values when taking the factorial of any number $n/2$ where $n$ is any integer greater than $-2$. Why does this happen? I thought factorials were only defined for positive integers and $0$, so what is my calculator doing to get the answer $3.32335097$ when I enter $2.5!$? Is there actually a definition of $2.5!$ or is my calculator just being weird? How is the factorial function implemented?
I understand the binary implications of $2.5$, so that could possibly have something to do with it. I get a domain error when trying to take the factorial of $-1, 2.3, e$, and any number that is not of the form $n/2$ where $n$ is any integer greater than $-2$.
Best Answer
There is an extension of the factorial to "most" numbers (including complex numbers) called the Gamma Function, $\Gamma(z) = \int_0^\infty e^{-t} t^{z-1}dt$. It satisfies $\Gamma(n+1) = n!$, and more generally, that $\Gamma(z+1) = z\Gamma(z)$ for any number $z$.
It is a curious fact that $\Gamma\left(\frac{1}{2}\right) = \sqrt{\pi}$. Using this, together with the fact that $\Gamma(z+1) = z\Gamma(z)$, we get
\begin{align*}2.5! &= \Gamma(3.5) \\\ &= 2.5\cdot\Gamma(2.5) \\\ &= 2.5\cdot 1.5\cdot \Gamma(1.5) \\\ &= 2.5\cdot 1.5\cdot .5\cdot \Gamma(.5)\\\ &= 2.5\cdot 1.5\cdot .5\cdot \sqrt{\pi} \\\ &= 3.32335097... \end{align*}