Laplace integrals

Laplace integrals are of the form:

where and are nice enough

Note that Watson’s lemma deals with the special case
(and , , so the endpoint dominates the integral)

To see a systematic way to find an Asymptotic Approximation
see Laplace Method
We give some examples below:

Principle of localisation

The Asymptotic Expansion of as is determined by
contributions from very small regions around the maxima of .

Case monotonic

Suppose is monotonic on
Let and let .
Then the integral is dominated by a nbd of :
Let and

Can use Watson’s lemma or just use partial integration:

If and

Case one local max

Suppose has one internal maximum at
(and this is the absolute maximum)
s.t. and and .

From the Case monotonic.

Then by noting that and

Now Taylor expand and .

Set

Now change limits in the integral
(later see that the induced error from this is very small):

Error induced by changing limits ?
Recall as
So error
So error is exponentially small

So so

Example

Taking doesn’t work because has no internal maxima.

Taking
we find max at
but this is not good because it depends on .

This motivates substitution ,
So

with max at

Example

If has one internal maximum and the same max at one of the endpoints.
Then the first order term contribution of the internal maximum dominates
because endpoint while internal is .

Example

If has many internal local maxima,
then only the absolute max matters.
For multiple absolute maximums, we just add them up.

Example

If has an absolute AND a local maximum at an endpoint,
the contribution is now

(this is done by doing the same manipulation as in Case one max
but the last integral is from to )

“Wide” maximum

Suppose

but

Proceed as in previous case but get:

Let

But the integral is just so

(the came from integrating only from to )

Higher order terms?

Plug in longer Taylor expansions …
wasn’t done in lectures, see
https://www.vle.cam.ac.uk/pluginfile.php/28347540/mod_resource/content/1/am_notes.pdf
Section 3.4.4

If then we NEED higher order terms