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.