Lecture 7: Saddle points

This lecture introduces the important concept of saddle points. By deforming the original contour of integration to pass though a saddle point in the complex plane, one can often obtain the full asymptotic expansion of an integral containing a large parameter x. The technique of saddle-point integration builds on the ideas in the past several lectures: Laplace's method (lecture 4), stationary phase (lecture 5), and steepest descent (lecture 6). Prof. Strogatz illustrates the method by using it to derive the first two terms in the asymptotic expansion for the Bessel function J_0(x) as x tends to infinity. This result improves on the result obtained by stationary phase, in which only the leading order term was obtained.