Nonlinear analysis with resurgent functions

TL;DR

This paper develops estimates for convolutions of resurgent functions, enabling nonlinear operations such as substitution, composition, and inversion, and justifies the rules of alien calculus within the framework of Borel-Laplace summability.

Contribution

It introduces new estimates for convolutions of resurgent functions, facilitating nonlinear operations and providing a rigorous foundation for alien calculus rules.

Findings

Estimates for convolution products of resurgent functions

Validation of nonlinear operations like substitution and inversion

Justification of alien calculus rules

Abstract

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.