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Automatic asymptotics. (English. French summary) Zbl 0874.65047
Paris: Univ. Paris VII, 405 p. (1997).
This thesis contributes to automatic computation in the infinitesimal calculus which occurs in the solution of systems of functional equations. It provides an algebraic theory which deals with formal series extended to “transseries”. Loosely speaking, a transseries is a series of which the elements are series which can be obtained, for instance, by using logarithm and exponential functions. As an example, $$f=\exp\{b\{\exp x\}\}$$ is a finite transseries, and $$f=\exp\{x^2+\exp\{\log_2^2 x+\exp\{\log_4^2 x+\dots\}$$ is an infinite transseries.
The book provides a rigorous formalism to analyze transseries which are strongly monotonic. Using this formalism, the author suggests a theoretical algorithm for solving algebraic differential equations by means of transseries. As a last contribution, one introduces multivariate transseries and transseries which are slowly oscillating.
The book is divided in two parts: Part A which deals with asymptotic algebra and Part B which involves automatic asymptotics. The main chapters of Part A are succesively: grid-based transseries, well-ordered transseries, the Newton polygon method, linear differential equations, algebraic differential equations, transvarieties. The main chapters of Part B read: asymptotic expansions of exp-log functions, automatic case separation, basics for automatic asymptotics, multivariate series, multivariate transseries, algebraic differential equations, oscillatory asymptotic behaviour. A sixty-pages Appendix provides the reader with some advanced prerequisites.
Unfortunately, this book suffers from a lack of illustrative detailed examples.

##### MSC:
 65L05 Numerical methods for initial value problems involving ordinary differential equations 34E05 Asymptotic expansions of solutions to ordinary differential equations 68W30 Symbolic computation and algebraic computation 34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.