Formal power series and linear systems of meromorphic ordinary differential equations.

*(English)*Zbl 0942.34004
Universitext. New York, NY: Springer. xviii, 299 p. (2000).

This book attempts to present the theory of linear differential equations in the complex domain, from the new perspective of multisummability. It briefly describes recent efforts on developing an analogous theory for nonlinear systems, systems of difference equations, partial differential equations and singular perturbation problems.

The book consists of 14 chapters and 3 appendices A, B, C. The chapter headings are as follows: 1. Basic properties of solutions; 2. Singularities of first kind; 3. Highest-level formal solutions; 4. Asymptotic power series; 5. Integral operators; 6. Summable power series; 7. Cauchy-Heine transform; 8. Solutions of highest-level; 9. Stokes’ phenomenon; 10. Multisummable power series; 11. Ecalle’s acceleration operator; 12. Other related questions; 13. Applications in other areas and computer algebra; 14. Some historical remarks; Appendix A. Matrices and vector spaces; B. Functions with values in Banach spaces; C. Functions of a matrix.

Each chapter is provided with exercises. Almost all these exercises are solved at the end of the book. In some exercises, the author gives specific suggestions, which are moderately challenging.

The book is well written and organized and is provided with a very nice preface, in which the author motivates the readers by several useful introductory examples to study this book.

The book consists of 14 chapters and 3 appendices A, B, C. The chapter headings are as follows: 1. Basic properties of solutions; 2. Singularities of first kind; 3. Highest-level formal solutions; 4. Asymptotic power series; 5. Integral operators; 6. Summable power series; 7. Cauchy-Heine transform; 8. Solutions of highest-level; 9. Stokes’ phenomenon; 10. Multisummable power series; 11. Ecalle’s acceleration operator; 12. Other related questions; 13. Applications in other areas and computer algebra; 14. Some historical remarks; Appendix A. Matrices and vector spaces; B. Functions with values in Banach spaces; C. Functions of a matrix.

Each chapter is provided with exercises. Almost all these exercises are solved at the end of the book. In some exercises, the author gives specific suggestions, which are moderately challenging.

The book is well written and organized and is provided with a very nice preface, in which the author motivates the readers by several useful introductory examples to study this book.

Reviewer: A.Klíč (Praha)

##### MSC:

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

30D30 | Meromorphic functions of one complex variable (general theory) |

34M05 | Entire and meromorphic solutions to ordinary differential equations in the complex domain |

30D10 | Representations of entire functions of one complex variable by series and integrals |

34E05 | Asymptotic expansions of solutions to ordinary differential equations |