Generalized analytic functions with applications to singular ordinary and partial differential equations

Bogdan Ziemian

GDML_Books, (1996), p.

Résumé

CONTENTS Introduction.............................................................................................................................................5 I. Preliminaries.........................................................................................................................................7    1. A review of classical results in the theory of Laplace integra............................................................7    2. Boundary values of holomorphic functions......................................................................................10      2.1. Distributions as boundary values of holomorphic functions.........................................................10      2.2. Hyperfunctions in one variable....................................................................................................12    3. Mellin analytic functionals, Mellin hyperfunctions and Mellin distributions.........................................14    4. Laplace distributions.........................................................................................................................18      4.1. Convolution of Laplace distributions.............................................................................................21    5. Ecalle distributions.............................................................................................................................23      5.1. Alien derivatives of Ecalle distributions.........................................................................................24    6. Paley-Wiener type theorem for Mellin analytic functionals.................................................................25      6.1. Phragmén-Lindelöf type theorems................................................................................................29    7. The cut-off functions and their Mellin transforms...............................................................................30    8. Modified Cauchy transformation in dimension 1.................................................................................31 II. The theory of generalized analytic functions..........................................................................................33    9. Definition of a generalized analytic function........................................................................................34    10. The Mellin transform of a generalized analytic function.....................................................................35    11. Characterization of GAFs in terms of Mellin transforms.....................................................................37    12. The Borel and Taylor transformations in the class of GAFs..............................................................40    13. Operations on generalized analytic functions...................................................................................40    14. Resurgent functions.........................................................................................................................44      14.1. Alien derivatives of resurgent functions......................................................................................46      14.2. Taylor-Fourier representation of resurgent functions..................................................................47 III. Applications to singular linear differential equations..............................................................................48    15. Special functions as generalized analytic functions...........................................................................48    16. Fuchsian type ODEs with generalized analytic coefficients................................................................52    17. Fuchsian type PDEs with "constant" coefficients................................................................................58    18. GAFs in several variables..................................................................................................................73    19. Fuchsian type PDEs with generalized analytic coefficients................................................................78 Appendices.................................................................................................................................................84 I. The symbol of a distribution in the sense of A. Weinstein. Conormal distributions...................................84 II. Nonlinear singular differential equations.................................................................................................88    1. The case of ordinary differential equations..........................................................................................88    2. The case of partial differential equations.............................................................................................93 References...................................................................................................................................................94 Symbol index.................................................................................................................................................97 Subject index................................................................................................................................................99

1991 Mathematics Subject Classification: 34A20, 35A20, 35C20, 46F12, 30D15, 46F15

Détail

EUDML-ID : urn:eudml:doc:270069