An Invitation to Q-series: From Jacobi's Triple Product Identity to Ramanujan's "most Beautiful Identity"

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World Scientific, 2011 - Mathematics - 226 pages
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The aim of these lecture notes is to provide a self-contained exposition of several fascinating formulas discovered by Srinivasa Ramanujan. Two central results in these notes are: (1) the evaluation of the Rogers–Ramanujan continued fraction — a result that convinced G H Hardy that Ramanujan was a “mathematician of the highest class”, and (2) what G. H. Hardy called Ramanujan's “Most Beautiful Identity”. This book covers a range of related results, such as several proofs of the famous Rogers–Ramanujan identities and a detailed account of Ramanujan's congruences. It also covers a range of techniques in q-series.
 

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Great book. I liked how everything it together.

Contents

Chapter 1 Introduction
1
Jacobis Triple Product Identity
3
The RogersRamanujan Identities
45
The RogersRamanujan Continued Fraction
95
From the Most Beautiful Identity to Ramanujans congruences
145
Appendix A Proofs of η 1τ iτ ητ
205
Appendix B The ranks of the partitions of n 9
209
Appendix C The cranks of the partitions of n 9
211
Bibliography
213
Index
225
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