Introduction to Quantum Mechanics

The saddle-point method is widely used in many physics problems for approximately evaluating integrals in which the integrand has one or a few sharp peaks. The basic idea is that the region around the sharp peak gives the dominant contribution to the integral. So, if we can find the contribution around the peak, we have an estimate of the integral. The reason for the name of the method will be explained below.

If f(z) has a sharp maxima, the main contribution to the integral would come from there, as we said in the prelude to this appendix. There are now several issues to worry about. First, the original contour may not pass through the stationary points. This is not a big concern because we are considering integrals for which the integrand is analytic everywhere. Therefore, the result should be independent of the path, because the integral of a differentiable complex function around any closed path is zero. We need to deform the contour C to another contour C′ that passes through the stationary points.

The second issue is that for differentiable complex functions, there is really no maxima. The Cauchy-Riemann conditions imply that both real and imaginary parts of a complex analytic function satisfy the 2D Laplace's equation. Therefore, if the second derivative is positive along one direction, it must be negative along the perpendicular direction. Thus, all stationary points are saddle points, and the method gets the name saddle-point method because of the association with these points.

So, passing the contour through the saddle point is not enough to ensure that we will obtain the dominant contribution to the integral. The deformed contour C′ must go through the direction along which the function really attains a maxima and falls sharply at points away from the maxima. Our estimate of the integral will be best if we find the path where the maxima is the steepest, and for this reason the method is also called the method of steepest descent.