Fuchsian Differential Equations: With Special Emphasis on the Gauss-Schwarz Theory PDF
Preview Fuchsian Differential Equations: With Special Emphasis on the Gauss-Schwarz Theory
Aspects of mathematics Masaaki Yoshida Fuchsian differential equations With special emphasis on the Gauss-Schwarz theory Masaaki Yoshida Fuchsian Differential Equations of Mathematics Asp3ds As~derMathematik Editor: Klas Diederich Vol. EJ: G. Hector/U. Hirsch, lntroduction to the Geometry of Foliatians, Part A Vol. E2: M. Knebusch/M. Kolster, Wittrings Vol. E3: G. Hector/U. Hirsch, lntraduction ta the Geometry of Foliations, Part B Vol. E4: M. Laska, Elliptic Curves over Number Fields with Prescribed Reduction Type Val. E5: P. Stiller, Automorphic Formsand the Picard Number of an Elliptic Surface Vol. E6: G. Faltings/G. Wüstholzet al., Rational Points (A Publication of the Max-Pianck-lnstitut für Mathematik, Bonn) Vol. E7: W. Stoll, Value Distribution Theory for Meromorph ic Maps Vol. E8: W. von Wahl, The Equations of Navier-Stokes and Abstract Parabolic Equations Val. E9: A. Haward, P.-M. Wong (Eds.), Contributians to Several Camplex Variables Vol. E10: A. J. Tromba, Seminar on New Results in Nonlinear Partial Differential Equations (A Publication of the Max-Pianck-lnstitut für Mathematik, Bonn) Vol. E11: M. Yoshida, Fuchsian Differential Equations (A Pub I ication of the Max-Pianck-1 nstitut für Mathematik, Bonn) Band D1: H. Kraft, Geometrische Methoden in der Invariantentheorie Masaaki Yoshida Fuchsian Differential Equations With Special Emphasis on the Gauss-Schwarz Theory A Publication of the Max-Pianck-lnstitut für Mathematik, Bann Adviser: Friedrich Hirzebruch Springer Fachmedien Wiesbaden GmbH ProfessorMasaaki Yoshida Kyushu University,Fukuoka,Japan AMSSubjectClassification:35R25, 35R30, 45A05,45L05,65F20 1987 All rights reserved © SpringerFachmedienWiesbaden1987 OriginallypublishedbyFriedr.Vieweg&SohnVerlagsgesellschaftmbH,Braunschweigin1987. Nopartof thispublication maybereproduced,storedin aretrieval system or transmitted in any form or by any means, electronic, mechanical,photocopying, recording or otherwise, without prior permissionofthe copyrightholder. Producedby W.Langelüddecke,Braunschweig ISSN 0179-2156 ISBN978-3-528-08971-9 ISBN978-3-663-14115-0(eBook) DOI 10.1007/978-3-663-14115-0 Contents Introduetion Notations Part I Chapter 1 Hypergeometrie Differential Equations ........... 1 § 1.1 Hypergeometrie Series ........................... 1 1.2 Hypergeometrie Equations ........................ 2 § 1.3 Contiguity Relations ............................ 3 1.4 Euler's Integral Representation ................. 5 1.5 Rarnes' Integral Representation ................ 11 1.6 Lonfluent Hypergeometrie Equations ............. 12 Chapter 2 General Theory of Differential Equations I .... 14 § 2.1 How to Write Differential Equations ............ 14 § 2. 2 Cauehy's Fundamental Theorem ................... 15 2. 3 Monodromy Representations of Differential Equations ....................................... 16 § 2. 4 Regular Singularities .......................... 18 2. 5 The Frobenius Method ........................... 20 § 2.6 Fuehsian Equations ............................. 24 Chapter 7 The Riemann and Riemann-Hilbert Problems ....... 27 J § 3.1 Statement of the Problems ...................... 27 3. 2 An Observation ................................. 28 3.3 The Solution of the Riemann Problem ............ 29 3.4 Apparent Singulari t ies ......................... 29 3. 5 A Solution of the Riemann-Hilbert Problem ...... 30 § 3.6 Isomonodromie Deformations ..................... 35 Chapter 4 Sehwarzian Derivatives I ...................... 38 4.1 Definitions and Properties ..................... 38 4.2 Relations With Differential Equations .......... 39 VI § 4.3 A Canonieal Form of Differential Equations ..... 42 § 4.4 Loeal Behaviour of Sehwarzian Derivative ....... 44 Chapter 5 The Gauss-Sehwarz Theory for Hypergeometrie Differential Equations ......................... 46 § 5. 1 Orbifolds and Their Uniformizations ............ 46 § 5. 2 Uniformizing Differential Equations ............ 48 § 5. 3 The Gauss-Sehwarz Theory ....................... 51 Part II Chapter 6 Hypergeometrie Differential Equations in Several Variables .............................. 58 6.1 Hypergeometrie Series .......................... 58 § 6.2 Hypergeometrie Differential Equations .......... 60 6.3 Contiguity Relations ........................... 63 6.4 Euler's Integral Representations .............. 65 § 6. 5 The Erdelyi-Takano Integral Representations .... 78 § 6.6 The Barnes Integral Representations ............ 78 § 6. 7 A Relation Between the Equation FD and the Garnier System GN ......................... 79 § 6.8 Confluent Hypergeometrie Equations ............. 81 Chapter 7 The General Theory of Differential Equations ... 82 7. 1 Singularities of Differential Equations ........ 82 § 7. 2 Holonomie Systems .............................. 85 § 7. 3 The Equation F1 ............................... 89 § 7.4 Equation' of Rank n+l .......................... 90 § 7. 5 Differential Equations on Manifolds ............ 92 § 7. 6 Regular Singularities .......................... 93 Chapter 8 Sehwarzian Derivatives II ..................... 95 § 8.1 Definitions and Properties ..................... 95 § 8. 2 Relations With Differential Equations .......... 99 § 8.3 A Canonieal Form of Differential Equations .... 100 § 8.4 Projeetive Equations and Projeetive Conneet ions ................................... 10 2 § 8. 5 Loeal Properties of Sehwarzian Derivatives .... 104 VII Chapter 9 The Riemann and Riemann-Hilbert Problems II .. 107 § 9.1 The Riemann Problem in Several Variables ...... 107 § 9. 2 Accessory Parameters .......................... 110 Chapter 10 The Gauss-Schwarz Theory in Two Variables ..... 115 § 10.1 Uniformizability of Orbifolds ................. 116 § 10.2 Orbifolds Whose Universal Uniformizations Are Symmetrie Spaces .......................... 118 § 10.3 Line Arrangements in P2 and Orbifolds Uniformized by B2 ............................ 124 10.4 Uniformizing Differential Equations ........... 135 § 10.5 The Gauss-Schwarz Theory for Appel1's Equation F1 .................................. 137 § 10.6 The Monodromy Representation of Appell's Equation F1 .................................. 145 Chapter 11 Reflection Groups ............................. 154 § 11. 1 Unitary Reflection Groups ..................... 155 § 11. 2 Unitary Reflection Groups of Dimension 2 ...... 159 11.3 Parabolic Reflection Groups of Dimension 2 .... 170 § 11.4 Line Arrangements Defined by Unitary Reflection Groups of Dimensions 3 and 4 ....... 179 Chapter 12 Toward Finding New Differential Equations ..... 181 § 12.1 Tensor Forms Associated to the Differential Equations ..................................... 181 § 12.2 Integrability Conditions ...................... 185 § 12.3 The Restricted Riemann Problem ................ 188 § 12.4 G-Invariant Differential Equations I ......... 191 12.5 G-Invariant Differential Equations II ........ 198 References ................................................. 204 Sources of Figures ......................................... 215 Relations of Chapters 2 ~ 3 4 1 ~~/ 5 7 ~ 9 6 11 / 12 "p + q" rneans "consu1t Chapter p when reading Chapter q" lntroduction This book stems from lectures given at the Max-Planck Institut fÜr Mathematik (Bonn) during the winter semester 1985/1986 under the title "Fuchsian Differential Equations". The aim of this series of lectures was to study linear ordinary differential equations and systems of linear partial differential equations with finite dimensional solution spaces in the complex analytic category; referred to in the course of this book as differential equations. This field has a long history and has attracted many farnaus mathematicians such as Euler, Gauss, Schwarz and Kummer, for example. Cauchy, L. and R. Fuchs, Painleve, Riemann, Poincare and Birkhoff were also interested in these equations. In many text books, as for instance in [Bie], in [Huk] and in [Inu], one can find the classical results in this direction, while the most recent ones are to be looked for in the book edited by Gerard and Okamoto [G-0]. The present work does not intend to be a new textbook discussing all aspects of the subject. On the contrary it concentrates exclusively on one particular but essential point, namely, the Gauss-Schwarz theory which deals with the hypergeometric differential equation. This theory is of special interest because its development was the occasion for many fruitful interactions between differential equations and other branches of mathematics, for example, algebraic geometry, group theory and differential geometry. For this reason, we hope that our book will give an opportunity to mathematicians working in other fields to get interested in differential equation theory -- which can undoubtedly be most profitable for their own work. The interrelations mentioned above originated partly in the age of Euler and Gauss, but have been rather overlooked in the recent past.
The list of books you might like
