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