Mathematical Surveys and Monographs 1991; 412 pp; hardcover Volume: 37 ISBN10: 0821816349 ISBN13: 9780821816349 List Price: US$154 Member Price: US$123.20 Order Code: SURV/37
 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 rapidlyexpanding field is brought up to date with two appendices, and the book concludes with an extensive bibliography of recent publications. About twothirds 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 transKummer 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  Kummertype 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
