NONLINEAR ANALYSIS WITH RESURGENT FUNCTIONS

We provide estimates for the convolution product of an arbitrary number of "resurgent functions", that is holomorphic germs at the origin of C that admit analytic continuation outside a closed discrete subset of C which is stable under addition. Such estimates are then used to perform nonlinear operations like substitution in a convergent series, composition or functional inversion with resurgent functions, and to justify the rules of "alien calculus"; they also yield implicitly defined resurgent functions. The same nonlinear operations can be performed in the framework of Borel-Laplace summability.