国外很好数学著作原版系列拉马努金笔记(第5卷)

Preface

Introduction

32 Continued Fractions

1 The Rogers-Ramanujan Continued Fraction

2 Other q-Continued Fractions

3 Continued Fractions Arising from Products of Gamma

Functions

4 Other Continued Fractions

5 General Theorems

33 Ramanuian’s Theories of Elliptic Functions to Alternative

Bases
Introduction Ramanujans Cubic Transformation, the borweins’ Cubic
Theta-Function Identity, and the Inversion Formula 3 The Principles of Triplication and Trimidiation

4 The Eisenstein Series L.M. and N

5 A Hypergeometric Transformation and Associated Transfer

Principle

6 More Higher Order Transformations for Hypergeometric

Series

7 Modular Equations in the Theory of Signature 3

8 The Inversion of an Analogue of k(k) in Signature 3

9 The Theory for Signature 4

10 Modular Equations in the Theory of Signature 4

11 The Theory for Signature 6

12 An Identity from the First Notebook and Further Hypergeometric Transformations
13 Some Enigmatic Formulas Near the End of the Third

Notebook

14 Concluding Remarks

34 Class Invariants and Singular Moduli

1 Introduction

2 Table of Class Invariants

3 Computation of Gn and gn when 9)n

4 Kronecker’s Limit Formula and General Formulas for Class

Invariants

5 Class Invariants Via Kronecker’s Limit Formula

6 Class Invariants Via Modular Equations

7 Class Invariants Via Class Field Theory

8 Miscellaneous results

9 Singular moduli

10 A Certain Rational Function of Singular Moduli

11 The Modular j-invariant

35 Values of Theta-Functions
Introduction Elementary values Nonelementary Values of p(e-n)
3 A Remarkable Product of Theta-Functions

36 Modular Equations and Theta-Function Identities in

Notebook 1

1 Modular Equations of degree 3 and Related Theta-Function

Identities

2 Modular Equations of Degree 5 and Related Theta-Function

Identities

3 Other Modular Equations and Related Theta-Function

Identities

4 Identities Involving Lambert Series

5 Identities Involving Eisenstein Series

6 Modular Equations in the Form of Schlafli

7 Modular Equations in the Form of Russell

8 Modular Equations in the Form of Weber

9 Series Transformations Associated with Theta-Functions

10 Miscellaneous Results

37 Infinite Series

38 Approximations and Asymptotic Expa
39 Miscellaneous Results in the First Notebook

Location of Entries in the Unorganized Portions of Ramanujan’s

First Notebook

References

lIndex

附录I

拉马努金的中国知音:数学家刘治国的“西天取经”之旅

附录Ⅱ

刘治国教授访谈

编辑手记