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Divergent series and asymptotic theories. Supplement of Bulletin de la Société Mathématique de France 121. (Séries divergentes et théories asymptotiques.) (French) Zbl 0830.34045
Panoramas et Synthèses 0. Paris: Société Mathématique de France. 74 p. (1993).
This monograph is a report on two talks given by the author on divergent series and asymptotic theories. It presents an introduction to classical results and recent works of the author and his coworkers on the subject. The first part, which is of historical character, explains on examples the interest of the subject. It points out the efficiency of the methods and prepares the necessary background for the theory. The second part deals with theoretical aspects. Results are clearly stated and the intuition is explained. No proofs are given; the author refers to the existing literature. The theory covers Gevrey asymptotic expansions, $$k$$- summability and the multisummability. A central result concerns a fundamental theorem for Gevrey asymptotic expansions which proves that formal series solutions of Gevrey type of differential equations are asymptotic in sectors to true solutions. The $$k$$-sommability complements such a result since it is related to the uniqueness of the sum of formal Gevrey series. The third part presents a fairly good idea of possible applications. This includes formal series of differential equations, normal form of differential equations and diffeomorphism, difference equations. Singular perturbations and ducks phenomena are presented using a nonstandard analysis approach. A possible interpretation of Evariste Galois’s last letter is discussed.

##### MSC:
 34E05 Asymptotic expansions of solutions to ordinary differential equations 40A30 Convergence and divergence of series and sequences of functions 34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations 40-02 Research exposition (monographs, survey articles) pertaining to sequences, series, summability