Expanding the factorial

What is the factorial of some $0 < x \leq 1$?

First what's a factorial? The factorial of a number n is the number of ways of arranging n objects in a line. Mathematically,

$n!=n(n-1)(n-2)(n-3)\dotsm(3)(2)(1)$

So $\begin{aligned}1! &= 1 \\ 2! &= 2 \\ 3! &= 6 \\ 4! &=24 \\ \dots \text{ and so on.} \end{aligned}$

So what is the value of $\dfrac{1}{2}!$?

To find the answer, let us first see the graph of $n!$

Desmos

Surprise! Surprise!! Turns out, $n!$ exists for negative integers as well. How? For that, we return to the graph again.

We can see that the function satisfies -

$\begin{aligned}f(1) &= 1 \\ f(x) &= xf(x)\end{aligned}$

It is a "smooth" graph. So $\ln f(x)$ is convex.

Turns out, there's actually another function that satisfies all the three above conditions - the gamma function.

$\Gamma(x)=\int_0^\infty \mathrm{t}^{x-1}{e}^{-t},\mathrm{d}t$

This would converge for some values of x, for some values, it won't.

So this leads to $x!=\Gamma(x+1)$

We can now find the value of $\dfrac{1}{2}! = \dfrac{\sqrt\pi}{2}$

Importance of the gamma function-

It pops up in many applications like quantum physics, fluid dynamics, statistics, number theory.

Further reading:

Bohr-Mollerup theorem

Numerical computation

Alternative ways

Inspiration