Defining $x!$ at fractions and when $x<0$

Question

Below is a screenshot from desmos(graph plotting app) . Here it shows that graph is continuous at all $x>0$ and tends to infinity at some values of $x$ and has very abrupt behaviour at $x<0$ . What I read till date is that it's only defined when $x$ is a whole number. Can someone explain what is it showing and how to calculate $x!$ at places except when $x$ is a whole number

Answer 1

This is (almost) the $\Gamma$ function. The $\Gamma$ function is defined as $$ \Gamma(x)=\int_0^\infty t^{x-1}e^{-t}dt $$ The $\Gamma$ function has, for positive integers $n$ (or really any complex numbers where it's defined) the properties $$ \Gamma(1)=1\\ \Gamma(n+1)=n\Gamma(n) $$ while the factorial has $$ 0!=1\\ (n+1)!=(n+1)\cdot n! $$ This is almost the same, but not quite. It can be remedied, however. Setting $f(x)=\Gamma(x+1)$ gives $$ f(0)=1\\ f(n+1)=(n+1)f(n) $$ which is to say $$ n!=f(n)=\Gamma(n+1) $$ And since the $\Gamma$ function is defined for any real (actually complex) apart from the non-positive integers, we can use this to extend the factorial to all real (actually complex) numbers apart from the negative integers. So what you see is the graph of $f$.

Note that the factorial itself is only defined for non-negative integers. The existence of the $\Gamma$ function doesn't change that.

The $\Gamma$ function is actually known to be the only analytic function on the positive real numbers which has $\Gamma(1)=1$, fulfills $\Gamma(x+1)=x\Gamma(x)$, and is superconvex (its logarithm is convex). In this sense it gives the nicest generalisation of the factorial to non-integers.