Formal power series and linear systems of meromorphic ordinary differential equations
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.
scientific article; zbMATH DE number 1391234