نظریه مدرن پایه‌ای معادلات تحلیلی پیچیده q-اختلافی

The roots of the modern theories of differential and q-difference equations go back in part to an article by George D. Birkhoff, published in 1913, dealing with the three “sister theories” of differential, difference and q-difference equations. This book is about q-difference equations and focuses on techniques inspired by differential equations, in line with Birkhoff's work, as revived over the last three decades. It follows the approach of the Ramis school, mixing algebraic and analytic methods. While it uses some q-calculus and is illustrated by q-special functions, these are not its main subjects. After a gentle historical introduction with emphasis on mathematics and a thorough study of basic problems such as elementary q-functions, elementary q-calculus, and low order equations, a detailed algebraic and analytic study of scalar equations is followed by the usual process of transforming them into systems and back again. The structural algebraic and analytic properties of systems are then described using q-difference modules (Newton polygon, filtration by the slopes). The final chapters deal with Fuchsian and irregular equations and systems, including their resolution, classification, Riemann-Hilbert correspondence, and Galois theory. Nine appendices complete the book and aim to help the reader by providing some fundamental yet not universally taught facts. There are 535 exercises of various styles and levels of difficulty. The main prerequisites are general algebra and analysis as taught in the first three years of university. The book will be of interest to expert and non-expert researchers as well as graduate students in mathematics and physics. Cover Title page Contents Foreword Preface Introduction 0.1. General orientation of this book 0.2. Contents 0.3. Some practical tips 0.4. General notations and conventions Chapter 1. Prelude 1.1. Fermat (Pierre de Fermat; French; first decade of 17th century–1665) 1.2. Euler (Leonhard Euler, Swiss, 1707–1783) 1.3. Gauß(Carl Friedrich Gauß, German, 1777–1855) 1.4. Cauchy (Augustin Louis Cauchy, French, 1789–1857) 1.5. Interlude: the q-exponentials 1.6. Heine (Eduard Heine, German, 1821–1881) 1.7. Jacobi (Carl Gustav Jakob Jacobi, German, 1804–1851) 1.8. Rogers (Leonard James Rogers, English, 1862–1933) 1.9. Jackson (Frank Hilton Jackson, English, 1870–1960) 1.10. Ramanujan (Srinivasa Ramanujan, Indian, 1887–1920) 1.11. Watson (George Neville Watson, English, 1886–1965) 1.12. The separation of q-calculus and \qde theory Chapter 2. Elementary Special and q-Special Functions 2.1. Classical Theta functions 2.2. “Basic” Theta functions 2.3. Application of Theta functions to the theory of \qdes 2.4. Degeneracies when q→1 Chapter 3. Basic Notions and Tools 3.1. Some q-difference algebra 3.2. Some preliminary generalities on solutions of \qdes 3.3. A bit of q-calculus Chapter 4. Equations of Low Order, Elementary Approach 4.1. Homogeneous equations of order 1 4.2. Nonhomogeneous equations of order 1 4.3. Homogeneous equations of order 2 4.4. Heine’s basic hypergeometric series 4.5. A first look at associated sheaves and bundles Chapter 5. Resolution of (General) Scalar Equations and Factorisation of q-Difference Operators 5.1. General considerations 5.2. Formal solutions and the Newton polygon 5.3. Adams’s lemma and analytic solutions 5.4. Factorisation 5.5. Application to the explicit resolution of scalar equations Chapter 6. Further Analytic Properties of Solutions: Index Theorems, Growth 6.1. Introductory examples 6.2. Index theorems and the growth of coefficients of divergent formal solutions (Bézivin) 6.3. Application to entire functions (Ramis) 6.4. A more elementary approach to index computations Chapter 7. Equations and Systems 7.1. From equations to systems and back 7.2. Generalities on systems 7.3. Sheaves of solutions Chapter 8. Systems and Modules 8.1. Generalities on \qdms 8.2. Morphisms of \qdms 8.3. Abelian properties of \qdms Chapter 9. Further Algebraic Properties of q-Difference Modules 9.1. Tensor properties 9.2. Homological properties 9.3. Sheaves of solutions of \qdms over \Ka Chapter 10. Newton Polygons and Slope Filtrations 10.1. Newton polygon 10.2. The canonical slope filtration 10.3. Graduation and formalization 10.4. Relation to other theories of slope filtrations Chapter 11. Fuchsian q-Difference Equations and Systems: Local Study 11.1. Local study 11.2. Confluence of fuchsian systems (local study) Chapter 12. Fuchsian q-Difference Equations and Systems: Global Study 12.1. Global study: the Riemann-Hilbert-Birkhoff correspondence for fuchsian systems 12.2. A geometric approach to Riemann-Hilbert-Birkhoff correspondence 12.3. Confluence again: global study Chapter 13. Galois Theory of Fuchsian Systems 13.1. Introduction to the Galois theory of \qdes 13.2. Local fuchsian Galois theory 13.3. Global fuchsian Galois theory Chapter 14. Irregular Equations 14.1. Introduction through an archetypal example 14.2. Irregular equations: discrete summation and asymptotics 14.3. A variant: continuous summation and asymptotics Chapter 15. Irregular Systems 15.1. The “additive” case of two slopes 15.2. The “general” case of k slopes: normal forms, basic tools 15.3. The Stokes phenomenon and the classification of irregular systems 15.4. The Galois group of irregular systems: a breezy account Appendix A. Some Classical Special Functions A.1. Elliptic functions and \El=\C/Λ A.2. Elliptic curves A.3. Modular functions A.4. The Gamma and the Zeta function A.5. Analyticity of roots Appendix B. Riemann Surfaces and Vector Bundles B.1. General definitions B.2. Functions B.3. Morphisms B.4. Differential forms (an informal presentation) B.5. General vector bundles B.6. Line bundles B.7. Analyticity of diagonalisation Appendix C. Classical Hypergeometric Functions C.1. The series and the equation C.2. Integral and connection formulas Appendix D. Basic Index Theory D.1. Indices in linear algebra D.2. Indices in functional analysis D.3. Dualities Appendix E. Cochain Complexes E.1. Cochain complexes and cohomology E.2. Morphisms of complexes, functoriality E.3. Homotopy Appendix F. Base Change and Tensor Products (and some more facts from linear algebra) F.1. Base change for vector spaces F.2. Tensor products of vector spaces F.3. Matricial view F.4. Some miscellaneous useful facts from linear algebra Appendix G. Tannaka Duality (without schemes) G.1. The weak form of Tannaka duality G.2. The strong form of Tannaka duality Appendix H. Čech Cohomology of Abelian Sheaves H.1. Coherent sheaves H.2. Čech cohomology H.3. Cohomology of algebraic coherent sheaves H.4. Some special sheaves Appendix I. Čech Cohomology of Nonabelian Sheaves I.1. Čech cohomology for sheaves of nonabelian groups I.2. Cohomology exact sequences I.3. Nonabelian cohomology and vector bundles Bibliography Index of terms Index of notations Index of names Back Cover