PH3203 : Lecture 10

This is the second part of the mathematical supplement on the use of the saddle point method. In this lecture I look at the case that is of direct relevance to us - that of the asymptotic behavior of Airy functions of the first and the second kind. I show how we can arrive at an integral representation for these functions, and then go on to use this integral representation to derive various properties. In particular I use the saddle point method to determine the asymptotics. This case takes a bit of care because of the presence of multiple saddle points in the problem. I try to explain in some detail how the notions of hills and valleys with respect to the saddle point is useful in choosing the actual path to be used. I finally end with an indication of how the integral transform method can be applied to other differential equations as well.