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

Ziemian Bogdan

  • Publisher: Instytut Matematyczny Polskiej Akademi Nauk(Warszawa), 1996

Abstract

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CONTENTSIntroduction.............................................................................................................................................5I. 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.................................................................................31II. 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..................................................................47III. 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................................................................78Appendices.................................................................................................................................................84I. The symbol of a distribution in the sense of A. Weinstein. Conormal distributions...................................84II. Nonlinear singular differential equations.................................................................................................88   1. The case of ordinary differential equations..........................................................................................88   2. The case of partial differential equations.............................................................................................93References...................................................................................................................................................94Symbol index.................................................................................................................................................97Subject index................................................................................................................................................991991 Mathematics Subject Classification: 34A20, 35A20, 35C20, 46F12, 30D15, 46F15

How to cite

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Ziemian Bogdan. Generalized analytic functions with applications to singular ordinary and partial differential equations. Warszawa: Instytut Matematyczny Polskiej Akademi Nauk, 1996. <http://eudml.org/doc/270069>.

@book{ZiemianBogdan1996,
abstract = {CONTENTSIntroduction.............................................................................................................................................5I. 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.................................................................................31II. 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..................................................................47III. 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................................................................78Appendices.................................................................................................................................................84I. The symbol of a distribution in the sense of A. Weinstein. Conormal distributions...................................84II. Nonlinear singular differential equations.................................................................................................88   1. The case of ordinary differential equations..........................................................................................88   2. The case of partial differential equations.............................................................................................93References...................................................................................................................................................94Symbol index.................................................................................................................................................97Subject index................................................................................................................................................991991 Mathematics Subject Classification: 34A20, 35A20, 35C20, 46F12, 30D15, 46F15},
author = {Ziemian Bogdan},
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},
location = {Warszawa},
publisher = {Instytut Matematyczny Polskiej Akademi Nauk},
title = {Generalized analytic functions with applications to singular ordinary and partial differential equations},
url = {http://eudml.org/doc/270069},
year = {1996},
}

TY - BOOK
AU - Ziemian Bogdan
TI - Generalized analytic functions with applications to singular ordinary and partial differential equations
PY - 1996
CY - Warszawa
PB - Instytut Matematyczny Polskiej Akademi Nauk
AB - CONTENTSIntroduction.............................................................................................................................................5I. 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.................................................................................31II. 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..................................................................47III. 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................................................................78Appendices.................................................................................................................................................84I. The symbol of a distribution in the sense of A. Weinstein. Conormal distributions...................................84II. Nonlinear singular differential equations.................................................................................................88   1. The case of ordinary differential equations..........................................................................................88   2. The case of partial differential equations.............................................................................................93References...................................................................................................................................................94Symbol index.................................................................................................................................................97Subject 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 -

Citations in EuDML Documents

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  1. Maria E. Pliś, Poincaré theorem and nonlinear PDE's
  2. M. E. Pliś, B. Ziemian, Borel resummation of formal solutions to nonlinear Laplace equations in 2 variables
  3. Grzegorz Łysik, Generalized analytic functions of Bogdan Ziemian
  4. Grzegorz Łysik, On 1-regular ordinary differential operators
  5. Grzegorz Łysik, Laplace integrals in partial differential equations in papers of Bogdan Ziemian
  6. Bogdan Bojarski, Stanisław Łojasiewicz, Grzegorz Łysik, Zofia Szmydt, Bogdan Ziemian (1953-1997)

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