Structural properties of polylogarithms:
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| Other Authors: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
[1991]
|
| Series: | Mathematical surveys and monographs
Volume 37 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xviii, 412 Seiten Diagramme |
| ISBN: | 0821816349 9780821816349 |
Staff View
MARC
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| 245 | 1 | 0 | |a Structural properties of polylogarithms |c Leonard Lewin, editor |
| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c [1991] | |
| 264 | 4 | |c © 1991 | |
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Record in the Search Index
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| adam_text | Contents
Preface xiii
Acknowledgments xv
List of Contributors xvii
Chapter 1. The Evolution of the Ladder Concept 1
L. Lewin
1.1 Early History 1
1.2 Functional Equations 2
1.3 More Recent Numerical Results 4
1.4 Current Developments 6
1.5 Base on the Unit Circle and Clausen Function Ladders 8
References 9
Chapter 2. Dilogarithmic Ladders 11
L. Lewin
2.1 Derivation from Kummer s Functional Equation 11
2.2 Relation to Clausen s Function 15
2.3 A Three Variable Dilogarithmic Functional Equation 17
2.4 Functional Equations in the Complex Plane 18
2.5 Cyclotomic Equations and Rogers Function 20
2.6 Accessible and Analytic Ladders 21
2.7 Inaccessible Ladders 23
References 25
Chapter 3. Polylogarithmic Ladders 27
M. Abouzahra and L. Lewin
3.1 Kummer s Function and its Relation to the Polylogarithm 27
3.2 Functional Equations for the Polylogarithm 28
3.3 A Generalization of Rogers Function to the nth Order 31
3.4 Ladder Order Independence on Reduction of Order 33
3.5 Generic Ladders for the Base Equation if + uq = 1 34
3.6 Examples of Ladders for n 3 40
3.7 Examples of Ladders for n 4 44
vii
viii CONTENTS
3.8 Examples of Ladders for n 5 45
3.9 Polynomial Relations for Ladders 46
References 47
Chapter 4. Ladders in the Trans Kummer Region 49
M. Abouzahra and L. Lewin
4.1 Ladder Results to n = 9 for the Base p 49
4.2 Ladder Results to n = 9 for the Base co 53
4.3 Ladder Results to n = 6 for the Base 8 62
4.4 The Nonexistence of Functional Equations at n = 6 with
Arguments Limited to ±zm(l z)r{ + z)s 65
References 67
Chapter 5. Supernumary Ladders 69
M. Abouzahra and L. Lewin
5.1 The Concept of Supernumary Results 69
5.2 Supernumary Results for p = 4 71
5.3 Supernumary Results for p = 5 76
5.4 Supernumary Results for p = 6 78
5.5 Supernumary Results for the Equation family
u6m+i + u6r~l = 1 80
5.6 Supernumary Results for an Irreducible Quintic 82
5.7 Supernumary Ladders from a 15 Term Functional Equation 84
5.8 Supernumary Ladders on the Unit Circle 90
References
Chapter 6. Functional Equations and Ladders 97
L. Lewin
6.1 New Categories of Functional Equations 97
6.2 The ^ family of Equations 100
6.3 The co family of Equations 109
6.4 The 0 family of Equations 115
Acknowledgements 121
References 121
Chapter 7. Multivariable Polylogarithm Identities 123
G. A. Ray
7.0 Introduction 123
7.1 A General Identity for the Dilogarithm 123
7.2 A General Identity for the Bloch Wigner Function 135
7.3 A General Identity for the Trilogarithm and Z 3(z) 141
7.4 Linear Power Relations among Dilogarithms 147
7.5 Cyclotomic Equations and Bases for Polylogarithm Relations 154
7.6 Mahler s Measure and Salem/Pisot Numbers 160
7.7 Recent Results for Supernumary Ladders 165
References 168
CONTENTS ix
Chapter 8. Functional Equations of Hyperlogarithms 171
G. Wechsung
8.1 Hyperlogarithms 171
8.2 Logarithmic Singularities 172
8.3 The Linear Spaces LIn and PLIn 176
8.4 Functional Equations of Hyperlogarithms 177
8.5 A Reduction Problem 181
References 184
Chapter 9. Kummer Type Functional Equations of Polylogarithms 185
G. Wechsung
9.1 Automorphic Functions 185
9.2 Kummer Type Functional Equations 186
9.3 A Method to Construct Functional Equations 191
9.4 The Nonexistence of a Kummer Type Functional Equation
for Li6 197
References 203
Chapter 10. The Basic Structure of Polylogarithmic Equations 205
Z. Wojtkowiak
10.1 Introduction 205
10.2 Canonical Unipotent Connection on P1(C) {a1, ... , an+1} 211
10.3 Horizontal Sections 213
10.4 Easy Lemmas about Monodromy 215
10.5 Functional Equations 216
10.6 Functional Equations of Polylogarithms 218
10.7 Functional Equations of Lower Degree Polylogarithms 223
10.8 Generalized Bloch Groups 228
Acknowledgements 231
References 231
Chapter 11. ^ Theory, Cyclotomic Equations and Clausen s Function 233
J. Browkin
11.1 Algebraic Background 233
11.2 Analytic Background 238
11.3 ^ theoretic Background 248
11.4 Examples 251
11.5 Problems and Conjectures 270
References 272
Chapter 12. Function Theory of Polylogarithms 275
S. Bloch
Chapter 13. Partition Identities and the Dilogarithm 287
J. H. Loxton
13.1 Introduction 287
x CONTENTS
13.2 Cyclotomic Equations 290
13.3 Accessible Relations 291
13.4 Partition Identities 292
13.5 Generalisations and Extensions 297
References 299
Chapter 14. The Dilogarithm and Volumes of Hyperbolic Polytopes 301
R. Kellerhals
14.0 Introduction 301
14.1 A Particular Class of Hyperbolic Polytopes 303
14.2 The Volume of a ^ Truncated Orthoscheme 309
14.3 Applications 321
14.4 Further Aspects 328
References 335
Chapter 15. Introduction to Higher Logarithms 337
R. M. Hain and R. MacPherson
15.1 The Problem of Generalizing the Logarithm and the
Dilogarithm 337
15.2 The Quest for Higher Logarithms 340
15.3 Higher Logarithms 341
15.4 The Higher Logarithm Bicomplex 343
15.5 Multivalued Deligne Cohomology 346
15.6 Higher Logarithms as Deligne Cohomology Classes 350
Acknowledgements 351
References 352
Chapter 16. Some Miscellaneous Results 355
L. Lewin
16.1 Clausen s Function and the Di Gamma Function for Rational
Arguments 355
16.2 An Infinite Integral of a Product of Two Polylogarithms 359
16.3 Cyclotomic and Polylogarithmic Equations for a Salem Number 364
16.4 New Functional Equations 373
References 374
Appendix A. Special Values and Functional Equations of
Polylogarithms 377
D. Zagier
0. Introduction 377
1. The Basic Algebraic Relation and the Definition of $fm(F) 378
2. Examples of Dilogarithm Relations 383
3. Examples for Higher Order Polylogarithms 385
4. Examples: Ladders 387
5. Existence of Relations among Polylogarithm Values of
Arbitrarily High Order 390
CONTENTS xi
6. A Conjecture on Linear Independence 391
7. Functional Equations 392
References 399
Appendix B. Summary of the Informal Polylogarithm Workshop,
November 17 18, 1990, MIT, Cambridge, Massachusetts 401
R. MacPherson and H. Sah
List of Participants 401
Abbreviated Summary 402
Bibliography 405
Index 409
|
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| spelling | Structural properties of polylogarithms Leonard Lewin, editor Providence, Rhode Island American Mathematical Society [1991] © 1991 xviii, 412 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Mathematical surveys and monographs Volume 37 Logarithmus (DE-588)4168047-9 gnd rswk-swf Polylogarithmische Funktion (DE-588)4301889-0 gnd rswk-swf Polylogarithmische Funktion (DE-588)4301889-0 s DE-604 Logarithmus (DE-588)4168047-9 s Lewin, Leonard 1927- (DE-588)1102913413 edt Erscheint auch als Online-Ausgabe 978-1-4704-1264-7 Mathematical surveys and monographs Volume 37 (DE-604)BV000018014 37 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002973372&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Structural properties of polylogarithms Mathematical surveys and monographs Logarithmus (DE-588)4168047-9 gnd Polylogarithmische Funktion (DE-588)4301889-0 gnd |
| subject_GND | (DE-588)4168047-9 (DE-588)4301889-0 |
| title | Structural properties of polylogarithms |
| title_auth | Structural properties of polylogarithms |
| title_exact_search | Structural properties of polylogarithms |
| title_full | Structural properties of polylogarithms Leonard Lewin, editor |
| title_fullStr | Structural properties of polylogarithms Leonard Lewin, editor |
| title_full_unstemmed | Structural properties of polylogarithms Leonard Lewin, editor |
| title_short | Structural properties of polylogarithms |
| title_sort | structural properties of polylogarithms |
| topic | Logarithmus (DE-588)4168047-9 gnd Polylogarithmische Funktion (DE-588)4301889-0 gnd |
| topic_facet | Logarithmus Polylogarithmische Funktion |
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