Ramanujan’s Notebooks, Part 5

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Springer Science & Business Media, Dec 12, 1997 - Mathematics - 624 pages
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During the years 1903-1914, Ramanujan recorded most of his mathematical dis coveries without proofs in notebooks. Although many of his results had already been published by others, most had not. Almost a decade after Ramanujan's death in 1920, G. N. Watson and B. M. Wilson began to edit Ramanujan's notebooks, but, despite devoting over ten years to this project, they never completed their task. An unedited photostat edition of the notebooks was published by the Tata Institute of Fundamental Research in Bombay in 1957. This book is the fifth and final volume devoted to the editing of Ramanujan's notebooks. Parts I-III, published, respectively, in 1985, 1989, and 1991, contain accounts of Chapters 1-21 in the second notebook, a revised enlarged edition of the first. Part IV, published in 1994, contains results from the 100 unorganized pages in the second notebook and the 33 unorganized pages comprising the third notebook. Also examined in Part IV are the 16 organized chapters in the first notebook, which contain very little that is not found in the second notebook. In this fifth volume, we examine the remaining contents from the 133 unorganized pages in the second and third notebooks, and the claims in the 198 unorganized pages of the first notebook that cannot be found in the succeeding notebooks.
 

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Contents

Continued Fractions
9
1 The RogersRamanujan Continued Fraction
12
2 Other qContinued Fractions
45
3 Continued Fractions Arising from Products of Gamma Functions
50
4 Other Continued Fractions
66
5 General Theorems
80
Ramanujans Theories of Elliptic Functions to Alternative Bases
89
2 Ramanujans Cubic Transformation the Borweins Cubic ThetaFunction Identity and the Inversion Formula
93
8 Miscellaneous Results
269
9 Singular Moduli
277
10 A Certain Rational Function of Singular Moduli
306
11 The Modular yinvariant
309
Values of ThetaFunctions
323
1 Elementary Values
325
2 Nonelementary Values of Peⁿ𝜋
327
3 A Remarkable Product of ThetaFunctions
337

3 The Principles of Triplication and Trimidiation
101
4 The Eisenstein Series L M and N
105
5 A Hypergeometric Transformation and Associated Transfer Principle
108
6 More Higher Order Transformations for Hypergeometric Series
116
7 Modular Equations in the Theory of Signature 3
120
8 The Inversion of an Analogue of Kk in Signature 3
133
9 The Theory for Signature 4
145
10 Modular Equations in the Theory of Signature 4
153
11 The Theory for Signature 6
161
12 An Identity from the First Notebook and Further Hypergeometric Transformations
165
13 Some Enigmatic Formulas Near the End of the Third Notebook
175
14 Concluding Remarks
180
Class Invariants and Singular Moduli
183
2 Table of Class Invariants
187
3 Computation of Gn and gn when 9n
204
4 Kroneckers Limit Formula and General Formulas for Class Invariants
216
5 Class Invariants Via Kroneckers Limit Formula
225
6 Class Invariants Via Modular Equations
243
7 Class Invariants Via Class Field Theory
257
Modular Equations and ThetaFunction Identities in Notebook 1
353
1 Modular Equations of Degree 3 and Related ThetaFunction Identities
354
2 Modular Equations of Degree 5 and Related ThetaFunction Identities
363
3 Other Modular Equations and Related ThetaFunction Identities
367
4 Identities Involving Lambert Series
373
5 Identities Involving Eisenstein Series
376
6 Modular Equations in the Form of Schlafli
378
7 Modular Equations in the Form of Russell
385
8 Modular Equations in the Form of Weber
391
9 Series Transformations Associated with ThetaFunctions
397
10 Miscellaneous Results
403
Infinite Series
409
Approximations and Asymptotic Expansions
503
Miscellaneous Results in the First Notebook
565
Location of Entries in the Unorganized Portions of Ramanujans First Notebook
579
References
605
Index
619
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