2.1 - Introduction to the Ecalle theory

Summary

Introduction to resurgence theory

This section is a very brief introduction to resurgence. Forgetting the purely technical difficulties, our aim is to present the noteworthy simple basic ideas of the Ecalle theory. In this way, we shall restrict ourselves to the quite simple algebra of simple resurgent functions, which gives a very pleasant context for beginning the theory.

The framework is the following. We begin by defining a subalgebra of the multiplicative algebra of formal power series C[[x^-1]], furnishing through the Borel transformation a convolutive subalgebra of analytic germs at the origin. On the other hand, in order to sum by a Laplace transformation, the analytic continuation of these germs must have only “few” singularities, a notion which has to be stable under the convolution product. After having defined the algebra of simple resurgent functions, we get naturally the notion of resurgent symbols by a comparison of the different summations, in other words by an analysis of the Stokes phenomena. These can be described either with the help of an automorphism of algebra or with new differentiations, the alien differentiations.

A bibliography will allow the reader to go further into the theory. At first, we naturally send the reader to the whole work of Ecalle himself. We have followed here more or less the clear presentation of the reference [CNP] where one can find all the basic tools with complete proofs and some applications.