Intuitive explanation for Factorial of negative fractional number How do you find out the factorial of negative fractional numbers. Does this make any sense? I don't understand. Can anybody give an intuitive explanation?
 A: On the non-negative integers, the factorial function is log-convex. That is, for $k\le m\le n$,
$$
\log(m!)\le\frac{m-k}{n-k}\log(n!)+\frac{n-m}{n-k}\log(k!)\tag{1}
$$
where
$$
m=\frac{m-k}{n-k}n+\frac{n-m}{n-k}k\tag{2}
$$
The Gamma function, defined as
$$
\Gamma(x)=\int_0^\infty t^{x-1}e^{-t}\,\mathrm{d}t\tag{3}
$$
is the unique log-convex function on $\mathbb{R}$ so that $\Gamma(1)=1$ and $x\Gamma(x)=\Gamma(x+1)$, which means that for a non-negative integer $n$,
$$
\Gamma(n+1)=n!\tag{4}
$$
The integral in $(3)$ is only convergent when $\mathrm{Re}(x)\gt0$, but because $x\Gamma(x)=\Gamma(x+1)$, we can extend the Gamma function to all of $\mathbb{C}$ except the non-positive integers.

Sometimes, due to $(4)$, people say $x!=\Gamma(x+1)$ for $x$ where factorial is not usually defined. In that case, we would have
$$
\left(-\frac12\right)!=\Gamma\left(\frac12\right)=\sqrt\pi\tag{5}
$$
A: the factorial function is only defined for non-negative integers. Because of this it makes no sense to speak about the factorial of negative fractional numbers. There are other functions that "extend" the factorial function to other domains, among these one of the most "Popular" is the Gamma function.
