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.

Reviews

"A remarkable book on a remarkable function, brimming with the enthusiasm of new discoveries and ideas for further research."

-- Zentralblatt MATH

Table of Contents

• L. Lewin -- The evolution of the ladder concept
• L. Lewin -- Dilogarithmic ladders
• M. Abouzahra and L. Lewin -- Polylogarithmic ladders
• M. Abouzahra and L. Lewin -- Ladders in the trans-Kummer region
• M. Abouzahra and L. Lewin -- Supernumary ladders
• L. Lewin -- Functional equations and ladders
• G. A. Ray -- Multivariable polylogarithm identities
• G. Wechsung -- Functional equations of hyperlogarithms
• G. Wechsung -- Kummer-type functional equations of polylogarithms
• Z. Wojtkowiak -- The basic structure of polylogarithmic functional equations
• J. Browkin -- \(K\)-theory, cyclotomic equations, and Clausen's function
• S. Bloch -- Function theory of polylogarithms
• J. H. Loxton -- Partition identities and the dilogarithm
• R. Kellerhals -- The dilogarithm and volumes of hyperbolic polytopes
• R. M. Hain and R. MacPherson -- Introduction to higher logarithms
• L. Lewin -- Some miscellaneous results
• D. Zagier -- Appendix A: Special values and functional equations of polylogarithms
• R. MacPherson and H. Sah -- Appendix B: Summary of the informal polylogarithm workshop