bash-4.2:~/posts/maths$ less stirling_formulae.html

I love Stirling's formula for the approximation of the factorial function :

        n
     ⎛n⎞      _________
n! ~ ⎜━⎟  ⋅ ╲╱2 ⋅ π ⋅ n
     ⎝e⎠

Note: The ~ means that both sides are asymptotic. Means that their ratio tends to 1 when n grows big.

It's actually been discovered first by Abraham de Moivre, and Stirling only found out that the constant in the formula was the square root of 2π.

Why I love it

First of all, it looks good.

But it also has something magical : it links many different mathematical concepts. The same formula contains : a factorial, a square root, π (pi), e (Euler's constant), an exponentiation...

It is also interesting for the relation it draws between factorial this weird exponentiation.

Behind the scenes, it is all about series and integrals, and also shows the growth rate of log(n!).