## Ramanujan’s Notebooks, Part 5During 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 |

605 | |

619 | |

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### Common terms and phrases

algebraic algebraic integers analytic continuation asymptotic expansion Bernoulli number Borweins Chan Chapter 19 Chapter 20 Chapter 34 class invariants coefficients complete the proof complex numbers Corollary cubic cusp Dedekind eta-function deduce defined definition Deleted by Ramanujan denotes Eisenstein series employ Entry 13 Entry 22 Entry 35 equality equation of degree equivalent Euler's gamma function Hence Jacobi triple product left side Lemma Mathematica modular equations modular forms Multiplying both sides nonnegative integer notation notebook 9 periodic continued fraction polynomials positive integer proof is complete proof of Entry proof of Theorem Ramanathan Ramanujan recorded Ramanujan's claim Ramanujan's notebooks Ramanujan's second notebook Recall Refs remains to show repulsive fixed point residue theorem respectively result right side root singular moduli suffices to prove table in Section table of Chapter tends to oo Theorem 5.6 theory of signature theta-function values Watson