Książka - szczegóły

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