Laplace Method

We have an integral of the form

and we want to find Asymptotic Approximation of
when

We assume that zeros and singularities of are not problematic.
If they are, we need to account for them separately.
Best way to do this is by some Watson’s lemma type argument.

The method

Draw picture of

We want to figure out the most important features of

Find the significant points

We want to find all the global maxima of .
This may include local maxima in
as well as the endpoints and

Truncation

We split the integral into small sections around the maxima

Expansion

For each maximum , write:

up to appropriate number of terms.
Note that we take and is the first nonzero term.
Crucially, is even (otherwise is a saddle, not a max)

Change variables

Use the change of variables:

and don’t forget also .
Note that the boundary now becomes large.
The exp bit of the integrand becomes:

Taylor expand

Use on terms past to get:

We now multiply this by the Taylor expansion of

Calculate the integrals

We first replace each boundary of form with .
This leaves us with integrals of form: