Structural Properties of Polylogarithms

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Leonard Lewin
American Mathematical Soc., 1991 - Mathematics - 412 pages
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Years ago, the handful of peculiar numerical dilogarithmic identities, known since the time of Euler and Landen, gave rise to new discoveries concerning cyclotomic equations and related polylogarithmic ladders. These discoveries were made mostly by the methods of classical analysis, with help from machine computation. About the same time, starting with Bloch's studies on the application of the dilogarithm in algebraic $K$-theory and algebraic geometry, many important discoveries were made in diverse areas. This book seeks to provide a synthesis of these two streams of thought. In addition to an account of ladders and their association with functional equations, the chapters include applications to volume calculations in Lobatchevsky geometry, relations to partition theory, connections with Clausen's function, new functional equations, and applications to $K$-theory and other branches of abstract algebra. This rapidly-expanding field is brought up to date with two appendices, and the book concludes with an extensive bibliography of recent publications. About two-thirds of the material is accessible to mathematicians and scientists in many areas, while the remainder requires more specialized background in abstract algebra.
  

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Contents

The Evolution of the Ladder Concept
1
Dilogarithmic Ladders
11
Polylogarithmic Ladders
27
Ladders in the TransKummer Region
49
Supernumary Ladders
69
Functional Equations and Ladders
97
Multivariable Polylogarithm Identities
123
Functional Equations of Hyperlogarithms
171
tfTheory Cyclotomic Equations and Clausens Function
233
Function Theory of Polylogarithms
275
Partition Identities and the Dilogarithm
287
The Dilogarithm and Volumes of Hyperbolic Polytopes
301
Introduction to Higher Logarithms
337
Some Miscellaneous Results
355
Appendix A Special Values and Functional Equations
377
Appendix B Summary of the Informal Polylogarithm Workshop
401

KummerType Functional Equations of Polylogarithms
185
The Basic Structure of Polylogarithmic Equations
205

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