Most asymptotic series arising in singularly perturbed problems diverge. How reliable, therefore, are predictions based on them, even if we generally focus on the first few terms? What do these divergences tell us? In the asymptotic context, divergence is the sign that exponentially small phenomena are hidden. An introductory example demonstrates this point in detail. Borel summation is introduced as a way to associate an objective and robust meaning of typical divergent series. A connection is made with the OGF and EGF of Chapter 6, which are indeed related to a Borel transform. An alternative technique to Borel summation, to which most of the chapter is devoted, is to truncate the series near its smallest term and study the remainder. With this technique, an original derivation of the – exponentially small – Stokes phenomenon associated with turning points with the WKB method is presented. A similar treatment is made on a third-order generalisation of Airy’s equation. Finally, the Stokes phenomenon is revealed in a multiple-scale problem. The interdependence between slow and fast scales is seen to be restored beyond all orders of the calculation.

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.