Ramanujan’s Notebooks: Part V, Partie 5Springer Science & Business Media, 6 déc. 2012 - 624 pages 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. |
Table des matières
1 | |
13 | |
Ramanujans Theories of Elliptic Functions to Alternative | 89 |
More Higher Order Transformations for Hypergeometric | 116 |
The Theory for Signature 4 | 145 |
Modular Equations in the Theory of Signature 4 | 153 |
Notebook | 175 |
Concluding Remarks | 180 |
Nonelementary Values of enя пл | 327 |
A Remarkable Product of ThetaFunctions | 337 |
Modular Equations and ThetaFunction Identities in Notebook 1 | 353 |
Modular Equations of Degree 3 and Related ThetaFunction Identities | 354 |
Modular Equations of Degree 5 and Related ThetaFunction Identities | 363 |
Other Modular Equations and Related ThetaFunction Identities | 367 |
Identities Involving Lambert Series | 373 |
Identities Involving Eisenstein Series | 376 |
Class Invariants and Singular Moduli 183 | 182 |
Table of Class Invariants | 187 |
Computation of Gn and gn when 9n | 204 |
Kroneckers Limit Formula and General Formulas for Class Invariants | 216 |
Class Invariants Via Kroneckers Limit Formula | 225 |
Class Invariants Via Modular Equations | 243 |
Class Invariants Via Class Field Theory | 257 |
Miscellaneous Results | 269 |
Singular Moduli | 277 |
A Certain Rational Function of Singular Moduli | 306 |
The Modular jinvariant | 309 |
Values of ThetaFunctions | 323 |
Elementary Values | 325 |
Modular Equations in the Form of Schläfli | 378 |
Modular Equations in the Form of Russell | 385 |
Modular Equations in the Form of Weber | 391 |
Series Transformations Associated with ThetaFunctions | 397 |
Miscellaneous Results | 403 |
Infinite Series 409 | 408 |
Approximations and Asymptotic Expansions | 503 |
Miscellaneous Results in the First Notebook | 565 |
Location of Entries in the Unorganized Portions of Ramanujans | 579 |
References | 605 |
619 | |
Autres éditions - Tout afficher
Expressions et termes fréquents
analytic continuation asymptotic expansion Borweins calculation Chapter 17 Chapter 20 Chapter 34 class invariants coefficients complete the proof complex numbers Corollary cubic deduce defined denotes Eisenstein series employ Entry 13 Entry 22 Entry 35 equality equation of degree equivalent F₁ follows formula function ƒ q³ ƒ³ gamma function given Hence identity Jacobi triple product left side Lemma log a log Math modular equations modular forms notation polynomials positive integer proof is complete proof of Entry proof of Theorem q¹² q¹³ q²)n Ramanujan Ramanujan's second notebook Refs replaced repulsive fixed point respectively result right side sin² ẞ has degree table in Section tends Theorem 8.1 theory of signature theta-function transformation values