# 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 $1!=1$

$2!=2$

$3!=6$

$4!=24$

$...$ and so on.

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

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 -

$f(1)=1 f(x)=xf(x)$

It is a "smooth" graph. So $lnf(x)$ 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 $\frac{1}{2}! = \frac{\sqrt\pi}{2}$

Importance of the gamma function-

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

Bohr-Mollerup theorem

Numerical computation

Alternative ways

Inspiration

2 weeks ago

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