PH3203 - Lecture 09

This lecture is a mathematical supplement which discusses a very useful technique for the estimation of the asymptotic behavior of a certain type of integrals. The specific kind of integral that is dealt with using this method is one where the integrand contains an exponential containing a large parameter. I start with Laplace's method for a estimation of asymptotic behavior of real integrals. After discussing this method with some stress on how to estimate corrections for large, but finite values of the controlling parameters, I illustrate its use with a couple of examples. I then go on to discuss in some detail the extension of this method to complex integrals. Here we have the advantage of being able to deform the contour to pass through a critical point of the function, but face the added complication that the real part of an analytic function can not have a maximum - but can have only a saddle point. I then go on to discuss the specific reason behind choosing a path of steepest descent for estimating the integral. I pay special attention to the geometry of the process - and talk about "hills" and "valleys" with respect to the saddle point. These concepts will be very useful in handling cases with multiple saddle points.