Method of Steepest Descent

We want to find Asymptotic Approximation of:

where is a curve in .
The size of the integrand is determined by
Write
We want to be small
Path of steepest descent is the one where decreases rapidly
So we look for the direction i.e. curves parallel to
Due to Cauchy-Riemann Equations

so we are looking for curves perpendicular to
But these are curves of constant .

The Method

Analyse

Separate

Stationary points

We find the stationary points of ,
similar to Laplace Method

Wells / valleys

We need places where as
We can freely deform the contour in here
as long as we stay in the same well/valley

Steepest descent contours

Find all the contours where
Note that these are parallel to
We specifically want the ones passing through
the end points of our integration contour
as well as the ones passing through stationary points of

Note that at stationary points, we will have two possible choices
We want the one that makes the real part go to

Deform

We want to deform into a combination of the above contours
such that it passes through some stationary points

Parametrise

Around each stationary point, parametrise
For lowest order, the first order approx is sufficient

Expand

After substituting our parametrisation,
evaluate the integral around each stationary point
similar to Laplace Method