The saddle point approximation

In this appendix we are going to see how the saddle point approximation works in general. Let us define the class of integrals:

Suppose that has a unique maximum at and that . Then, expanding around we have:
(and obviously because is a maximum). Setting we can write, stopping the expansion at the third order:
Calling and remembering that since is a maximum, we have:
Therefore for very large the term proportional to in the exponential (like all the following terms of the complete expansion) is negligible, so:
and computing the Gaussian integral:
Therefore we see that the saddle point approximation essentially states that an integral of the form can be approximated, provided that is large, with the value of the integrand calculated at its maximum (up to a multiplicative factor).