Stirling’s formula says

on means that the ratio of the two quantities goes to how

Where does this formula come from? In particular, how the name works participate in it? To understand these things, I think a non-rigorous argument that can be made rigorous is more useful than a rigorous test with all the i points and t crossed. I think it’s important to keep the argument *short*. So let me do this. The punchline will be that the comes from this formula:

And that, I hope you know, comes from squaring the two sides and turning the left side into an integral integral that you can do in polar coordinates, taking out a factor of because it only depends on what you are integrating no

Okay, here it goes. Let’s start with

It can be easily shown by repeated integration by parts.

Then we do this:

In the first step, we are writing how In the second we change the variables:

Then we can use to ruin things:

All the hard work will be done proving this:

With that in mind, we get it

and simplifying we obtain the formulas of Stirling:

### Laplace method

How do we achieve this?

Let’s write it like

The trick is to watch it as becomes large, the integral will be dominated by the point at which is as small as possible. Then we can approximate the integral for a Gaussian with a maximum point at this time!

Realize that

thus the function has a critical point a and its second derivative is there, so it’s a local minimum. In fact, this point is a minimum throughout the range

Then we use this:

**Laplace method.** Suppose has a unique minimum at some point i 0.” class=”latex”/> Then

how

This is demonstrated by approximating the integral by a Gaussian. Applying it to our case at hand, we succeed

on $ f (y_0) = 1 $ i $ f ”(y_0) = 1. $ So we get

and then leaving it we get what we want.

Therefore, from this point of view — and there are others — the key to Stirling’s formula is Laplace’s method of approximating an integral as

with a Gaussian integral as In the end, the crucial calculation is

You can see the full proof of Laplace’s method here:

• Wikipedia, Laplace method.

Physicists who have made quantum field theory will know that when the thrust is shown it is mostly Gaussian integrals: the limit what we’re seeing here is like a “classic limit” where Therefore, they will know this idea.

Here there should be a deeper morality, about how it is related to some Gaussian process, but I do not know, although I know how binomial coefficients approximate a Gaussian distribution. Know some deeper explanation, perhaps in terms of probability theory and combinatorics, of why ends up being described asymptotically by an integral of a Gaussian?

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