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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 ¸½Â¼I À­ÂíŬ½ðµÄÖйúÖªÒô:Êýѧ¼ÒÁõÖιúµÄ“Î÷ÌìÈ¡¾­”Ö®Âà ¸½Â¼¢ò ÁõÖιú½ÌÊÚ·Ã̸ ±à¼­ÊÖ¼Ç