The WKB method is the resolution method of choice for singularly perturbed linear ordinary differential equations. It is introduced by the historical example of its first user, Francesco Carlini, in his treatment of Kepler’s two-body problem in 1817. At that stage, the presentation of the method is not yet complete, but one is already capable of solving boundary layer problems and wave and vibrations problems and discussing Berry’s phase in quantum mechanics. Then, the presence of singularities, known as turning points, is explained and treated in a number of applications ranging from quantum mechanics to fluid instabilities. On either side of a turning point, some constants of integration change value, a mathematical behaviour called the Stokes phenomenon. We discuss it in detail and relate this effect to the divergence of the asymptotic series generated by the WKB method. This prepares the reader for Chapter 7 of the book. Second-order turning points and an alternative implementation of the method based on the Airy function are also presented.

This chapter discusses the principal methods of approximations of integrals: integration by parts, Laplace, saddle point, and stationary phase. Abundant use is made of numerical comparisons. In particular, the saddle-point method, a.k.a. method of steepest descent, is introduced by a couple of numerical experiments with Mathematica in which the path of integration is discretised and deformed, in application of Cauchy’s theorem. Among the examples treated in this chapter, the saddle-point method is applied, step by step, to the evaluation of the Airy function and a generalisation thereof, namely the integral representation of hyper-Airy functions.