In this appendix we are going to see how the saddle point approximation works in general.
Let us define the class of integrals:
has a unique maximum at
. Then, expanding
is a maximum). Setting
we can write, stopping the expansion at the third order:
and remembering that
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).