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Paralogarithms constitute a family of special functions, which are somegeneralizations of hyperlogarithms. They have been introduced by Jean Ecalle inthe context of the classification of complex analytic dynamical systems withirregular singularities, to solve the so-called "synthesis problem" in aneffective and very general way. We describe the formalism of resurgencemonomials, introduce the paralogarithmic family and present the effectivesynthesis with paralogarithmic monomials, of analytic vector fields having asaddle-node singularity, following Ecalle.

Ecalle's paralogarithmic resurgence monomials and effective synthesis