Produktdetaljer
ISBN
9781441926661
Publisert
2010-10-29
Utgiver
Vendor
Springer-Verlag New York Inc.
Høyde
235 mm
Bredde
155 mm
Aldersnivå
Research, P, 06
Språk
Product language
Engelsk
Format
Product format
Heftet
From the reviews: "This volume contains 16 chapters comprising 314 entries. The material is arranged thematically with the main topics being some of Ramanujan's favorites q series theta functions ... . the authors treatment is extremely thorough. Each chapter contains an introduction with appropriate background. References to all other known proofs of the entries are provided. ... Fans of Ramanujan's mathematics are sure to be delighted by this book. ... Many entries are just begging for further study and will undoubtedly be inspiring research for decades to come." (Jeremy Lovejoy, Mathematical Reviews, Issue 2010 f)
This is the second of approximately four volumes that the authors plan to write in their examination of all the claims made by S. Ramanujan in The Lost Notebook and Other Unpublished Papers. This volume, published by Narosa in 1988, contains the “Lost Notebook,” which was discovered by the ?rst author in the spring of 1976 at the library of Trinity College, Cambridge. Also included in this publication are other partial manuscripts, fragments, and letters that Ramanujan wrote to G. H. Hardy from nursing homes during 1917–1919. The authors have attempted to organize this disparate material in chapters. This second volume contains 16 chapters comprising 314 entries, including some duplications and examples, with chapter totals ranging from a high of ?fty-four entries in Chapter 1 to a low of two entries in Chapter 12. Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Introduction . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 The Heine Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1. 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1. 2 Heine’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1. 3 Ramanujan’s Proof of the q-Gauss Summation Theorem . . . . . 10 1. 4 Corollaries of (1. 2. 1) and (1. 2. 5) . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1. 5 Corollaries of (1. 2. 6) and (1. 2. 7) . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1. 6 Corollaries of (1. 2. 8), (1. 2. 9), and (1. 2. 10) . . . . . . . . . . . . . . . . . . 24 1. 7 Corollaries of Section 1. 2 and Auxiliary Results . . . . . . . . . . . . . 27 2 The Sears–Thomae Transformation . . . . . . . . . . . . . . . . . . . . . . . . 45 2. 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 45 2. 2 Direct Corollaries of (2. 1. 1) and (2. 1. 3) . . . . . . . . . . . . . . . . . . . . 45 2. 3 Extended Corollaries of (2. 1. 1) and (2. 1. 3) . . . . . . . . . . . . . . . . .
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Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Introduction . 1 1 The Heine Transformation . 1 Introduction . 2 Heine’s Method . 3 Ramanujan’s Proof of the q-Gauss Summation Theorem . 5) . 7) . 10) . 2 and Auxiliary Results . 27 2 The Sears–Thomae Transformation . 1 Introduction . 3) . 3) .
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The Heine Transformation.- The Sears#x2013; Thomae Transformation.- Bilateral Series.- Well-Poised Series.- Bailey#x02019;s Lemma and Theta Expansions.- Partial Theta Functions.- Special Identities.- Theta Function Identities.- Ramanujan#x02019;s Cubic Analogue of the Classical Ramanujan#x2013;Weber Class Invariants.- Miscellaneous Results on Elliptic Functions and Theta Functions.- Formulas for the Power Series Coefficients of Certain Quotients of Eisenstein Series.- Two Letters on Eisenstein Series Written from Matlock House.- Eisenstein Series and Modular Equations.- Series Representable in Terms of Eisenstein Series.- Eisenstein Series and Approximations to #x03C0;.- Miscellaneous Results on Eisenstein Series.
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This volume is the second of approximately four volumes that the authors plan to write on Ramanujan’s lost notebook, which is broadly interpreted to include all material published in The Lost Notebook and Other Unpublished Papers in 1988. The primary topics addressed in the authors’ second volume on the lost notebook are q-series, Eisenstein series, and theta functions. Most of the entries on q-series are located in the heart of the original lost notebook, while the entries on Eisenstein series are either scattered in the lost notebook or are found in letters that Ramanujan wrote to G.H. Hardy from nursing homes. About Ramanujan's Lost Notebook, Volume I: "Andrews and Berndt are to be congratulated on the job they are doing. This is the first step...on the way to an understanding of the work of the genius Ramanujan. It should act as an inspiration to future generations of mathematicians to tackle a job that will never be complete." - Gazette of the Australian Mathematical Society "...the results are organized topically with cross-references to the identities as they appear in the original Ramanujan manuscript. Particularly helpful are the extensive references, indicating where in the literature these results have been proven or independently discovered as well as where and how they have been used." - Bulletin of the American Mathematical Society "The mathematics community owes a huge debt of gratitude to Andrews and Berndt for undertaking the monumental task of producing a coherent presentation along with complete proofs of the chaotically written mathematical thoughts of Ramanujan during the last year of his life. Some 85 years after his death, beautiful "new" and useful results of Ramanujan continue to be brought to light." - Mathematical Reviews
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From the reviews: "This volume contains 16 chapters comprising 314 entries. The material is arranged thematically with the main topics being some of Ramanujan's favorites q series theta functions ... . the authors treatment is extremely thorough. Each chapter contains an introduction with appropriate background. References to all other known proofs of the entries are provided. ... Fans of Ramanujan's mathematics are sure to be delighted by this book. ... Many entries are just begging for further study and will undoubtedly be inspiring research for decades to come." (Jeremy Lovejoy, Mathematical Reviews, Issue 2010 f)
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Most of this material has never before been published in book form Authors have organized, and provided commentary on, Ramanujan's results Discusses q-series, Eisenstein series, and theta functions Includes letters to G.H. Hardy
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Produktdetaljer
ISBN
9781441926661
Publisert
2010-10-29
Utgiver
Vendor
Springer-Verlag New York Inc.
Høyde
235 mm
Bredde
155 mm
Aldersnivå
Research, P, 06
Språk
Product language
Engelsk
Format
Product format
Heftet