拉馬努金筆記（英文版 第5卷）

印度神奇數學家拉馬努金有著深邃的數學直覺和洞察力，這套叢書給出了拉馬努金很多漂亮的數學結果，共分為5卷。本書為第5卷，主要介紹了數論、素數的拉馬努金理論、模方程的函數、雙紐線與聯合方程的逆公式等內容。本書適合高年級本科生、研究生，以及數學愛好者參考閱讀。

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 Ramanujan's Theories of Elliptic Functions to Alternative Bases
1 Introduction
2 Ramanujan's 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 9In
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-Funetions
0 Introduction
1 Elementary Values
2 Nonelementary Values of φo(e-nπ)
3 A Remarkable Product of Theta-Functions
36 Modular Equations and Theta-Funetion 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