The Method of Stationary Phase

Consider the Oscillatory Integrals:

where is stationary at a point .

Method

Suppose that
It is very similar if is of higher order.

Taylor expand

Write:

Separate into sections

Let be small (but will depend on )
By Oscillatory Integrals with monotonic phase,
we know that in the areas of monotonic
we have contribution.
Thus we can write:

Use substitution

Substitute

We should be left with an integral of form

which we solve using The Fresnel Integrals.
Note that we did have to choose a suitable for this to work,
but the exact doesn’t really matter.