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

Bogdan Ziemian. Generalized analytic functions with applications to singular ordinary and partial differential equations. 1996. <http://eudml.org/doc/270069>.

@book{BogdanZiemian1996,
abstract = {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................................................................................................................................................991991 Mathematics Subject Classification: 34A20, 35A20, 35C20, 46F12, 30D15, 46F15},
author = {Bogdan Ziemian},
keywords = {generalized analytic functions; GAF-functions; Laplace transformation of generalized functions; Mellin analytic functionals; Borel and Taylor transformations; change of variables; Fuchsian type ordinary and partial differential equations},
language = {eng},
title = {Generalized analytic functions with applications to singular ordinary and partial differential equations},
url = {http://eudml.org/doc/270069},
year = {1996},
}

TY - BOOK
AU - Bogdan Ziemian
TI - Generalized analytic functions with applications to singular ordinary and partial differential equations
PY - 1996
AB - 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................................................................................................................................................991991 Mathematics Subject Classification: 34A20, 35A20, 35C20, 46F12, 30D15, 46F15
LA - eng
KW - generalized analytic functions; GAF-functions; Laplace transformation of generalized functions; Mellin analytic functionals; Borel and Taylor transformations; change of variables; Fuchsian type ordinary and partial differential equations
UR - http://eudml.org/doc/270069
ER -