The method of stationary phase gives us a quick and easy way to asymptotically approximate an integral whose integrand is a rapidly oscillating function, in the limit as the frequency of the oscillations tends to infinity. This is one of the oldest asymptotic methods, having been developed by Gabriel Stokes and Lord Kelvin in the 1800s. Today the method of stationary phase is used throughout science, mathematics, and engineering, in fields such as optics, fluid dynamics, electromagnetism, quantum mechanics, and mathematical biology. In this lecture, Prof. Strogatz introduces the method and uses it to approximate the large-x behavior of one of the most famous special functions, the Bessel function J_0(x). As a bonus, the end of the lecture shows how to use the complex analysis technique known as contour integration to calculate the Fresnel integrals, i.e., the definite integrals of sin(x^2) and cos(x^2) from 0 to infinity, which arise in the method of stationary phase.

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