Stirling's approximation

We want to derive here Stirling's approximation, i.e. we want to show that for large we have:

We start by writing with Euler's function:

Changing variable to we get:
which we rewrite as:
The last one is an integral in the form of those studied in the appendix The saddle point approximation; using the notation shown there we have and its maximum is at . Since , we get:

Taking the logarithm we find another famous expression for Stirling's approximation: