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Nonlinear analysis with resurgent functions. (Analyse non linéaire pour les fonctions résurgentes.) (English. French summary) Zbl 1326.30036

Summary: We provide estimates for the convolution product of an arbitrary number of “resurgent functions”, that is holomorphic germs at the origin of \(\mathbb C\) that admit analytic continuation outside a closed discrete subset of \(\mathbb C\) 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.

MSC:

30E99 Miscellaneous topics of analysis in the complex plane
30B40 Analytic continuation of functions of one complex variable
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