Table Of ContentAspects 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
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Val. E5: P. Stiller, Automorphic Formsand the Picard
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(A Publication of the Max-Pianck-lnstitut für Mathematik, Bonn)
Vol. E7: W. Stoll, Value Distribution Theory for
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Vol. E8: W. von Wahl, The Equations of Navier-Stokes
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Val. E9: A. Haward, P.-M. Wong (Eds.), Contributians
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Vol. E10: A. J. Tromba, Seminar on New Results in Nonlinear
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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.
