Ecalle's paralogarithmic resurgence monomials and effective synthesis

Abstract

Paralogarithms constitute a family of special functions, which are some generalizations of hyperlogarithms. They have been introduced by Jean Ecalle in the context of the classification of complex analytic dynamical systems with irregular singularities, to solve the so-called "synthesis problem" in an effective and very general way. We describe the formalism of resurgence monomials, introduce the paralogarithmic family and present the effective synthesis with paralogarithmic monomials, of analytic vector fields having a saddle-node singularity, following Ecalle.

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