Expanding the factorial

What is the factorial of some 0<x10 < 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(n1)(n2)(n3)(3)(2)(1)n!=n(n-1)(n-2)(n-3)\dotsm(3)(2)(1)

So 1!=12!=23!=64!=24 and so on.\begin{aligned}1! &= 1 \\ 2! &= 2 \\ 3! &= 6 \\ 4! &=24 \\ \dots \text{ and so on.} \end{aligned}

So what is the value of 12!\dfrac{1}{2}!?

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

Desmos Desmos

Surprise! Surprise!! Turns out, n!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)=1f(x)=xf(x)\begin{aligned}f(1) &= 1 \\ f(x) &= xf(x)\end{aligned}

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

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

Γ(x)=0tx1et,dt\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!=Γ(x+1)x!=\Gamma(x+1)

We can now find the value of 12!=π2\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

Note by Adhiraj Dutta
6 months ago

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Really good! But calculators don't show it, why? @Adhiraj Dutta

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Because they don't use the gamma function. Instead they use an approximation using Stirling's approximation.

I found this on a reddit post.

Adhiraj Dutta - 3 months ago

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I have a challenge for you - can you find π4\frac{\pi}{4} !!?

A Former Brilliant Member - 2 months, 1 week ago

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Isn't it n=0π3((n+1)(n+3))?\displaystyle \sum_{n = 0}^\infty \frac{\pi}{3((n+1)(n+3))} ? :P

Adhiraj Dutta - 2 months, 1 week ago

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That's not what my question was about...

A Former Brilliant Member - 2 months, 1 week ago

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@A Former Brilliant Member Did you mean factorial of π4?\dfrac{\pi}{4}?. Use Wolfram Alpha.

Adhiraj Dutta - 2 months, 1 week ago

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