Introduction to 1-summability and resurgence

Abstract: This text is about the mathematical use of certain divergent power series. The first part is an introduction to 1-summability. The definitions rely on the formal Borel transform and the Laplace transform along an arbitrary direction of the complex plane. Given an arc of directions, if a power series is 1-summable in that arc, then one can attach to it a Borel-Laplace sum, i.e. a holomorphic function defined in a large enough sector and asymptotic to that power series in Gevrey sense. The second part is an introduction to Ecalle's resurgence theory. A power series is said to be resurgent when its Borel transform is convergent and has good analytic continuation properties: there may be singularities but they must be isolated. The analysis of these singularities, through the so-called alien calculus, allows one to compare the various Borel-Laplace sums attached to the same resurgent 1-summable series.In the context of analytic difference-or-differential equations, this sheds light on the Stokes phenomenon. A few elementary or classical examples are given a thorough treatment (the Euler series, the Stirling series, a less known example by Poincar\'e). Special attention is devoted to non-linear operations: 1-summable series as well as resurgent series are shown to form algebras which are stable by composition. As an application, the resurgent approach to the classification of tangent-to-identity germs of holomorphic diffeomorphisms in the simplest case is included. An example of a class of non-linear differential equations giving rise to resurgent solutions is also presented. The exposition is as self-contained as can be, requiring only some familiarity with holomorphic functions of one complex variable.