国外很好数学著作原版系列拉马努金遗失笔记(第1卷)

Introduction 1 The Rogers-Ramanujan Continued Fraction and Its Modular Properties 1.1 Introduction 1.2 Two-Variable Generalizations of (1.1.10) and(1. 1.11) 13 1.3 Hybrids of(11.10)and(1.1.11) 1.4 Factorizations of(1.1.10) and(1. 1.11) 1.5 Modular equations 1.6 Theta-Function Identities of Degree 5 1.7 Refinements of the Previous Identities 1.8 Identities Involving the Parameter k=R(q)R(q2) 1.9 Other Representations of Theta Functions Involving R(q)..39 1.10 Explicit Formulas Arising from(1.1.11)….……,44 2 Explicit Evaluations of the Rogers-Ramanujan Continued Fraction 2.1 Introduction 2.2 Explicit Evaluations Using Eta-Function Identities 2.3 General Formulas for Evaluating R(e-2mVn) and S(e-TVn).66 2.4 Page 210 of Ramanujan's Lost Notebook 2.5 Some Theta-Function Identities 2.6 Ramanujans General Explicit Formulas for the Rogers-Ramanujan Continued Fraction 79 3 A Fragment on the Rogers-Ramanujan and Cubic Continued fractions 3.1 Introduction 3.23 The RogersTheory-RamanujofanujaContinuedsCubicFractionContinued Fraction,...86 3.4 Explicit Evaluations of G(a) 4 Rogers-Ramanujan Continued Fraction- Partitions, Lambert series 4.1 Introduction.....,,,,.,........... 4.2 Connections with Partitions 4.3 Further Identities Involving the Power Series Coefficients of C(q)and1/C(q)…… 4.4 Generalized Lambert Series 4.5 Further g-Series Representations for C(a) 5 Finite Rogers-Ramanujan Continued Fractions......125 5.1 Introduction......... 5.2 Finite Rogers-Ramanujan Continued Fractions......126 53 A generalization of Entry5.2.1..………∵ 5.4 Class invariant 5.5 A Finite Generalized Rogers-Ramanujan Continued Fraction 140 6 Other q-continued fractions 6.1 Introduction 6.2The Main Theore 6.3A Second General Continued Fraction 6.4 A Third General Continued Fraction...........1596.5 A Transformation Formula 6.6 Zeros................,165 6.7 Two Entries on Page 200 of Ramanujan's Lost Notebook.. 169 6.8 An Elementary Continued Fraction 7 Asymptotic Formulas for Continued Fractions 7.1 Introduction 7.2 The Main Theorem 7.3 Two Asymptotic Formulas Found on Page 45 of Ramanujans Lost Notebook 7.4 An Asymptotic Formula for R(a, q) 8 Ramanujan,s Continued Fraction for(q 8.1 Introduction 8.2 A Proof of Ramanujan's Formula(8.1.2) 3 The Special Case a= w of(8.1.2)8.4 Two Continued Fractions Related to(q; q)oo/(q; oo... 213 8.5 An Asymptotic Expansion 9 The Rogers-Fine Identity 1 Introduction........ 9.2Series Transformations 9.3The Seriesnan(n 1)/2 n=09n(3n 1)/2 9. 4 The Series 9.5 The Seriesn=o gun 2n 10 An Empirical Study of the Rogers-Ramanujan Identities. 241 10.1 Introduction.......,,,,,∴,.241 10.2 The First Argument 10.3 The Second Argument 10.4 The Third Argument 10.5 The Fourth Argument 11 Rogers-Ramanujan-Slater-Type Identities........ 251 11.1 Introduction. 11.2 Identities Associated with Modulus 5.,.................. 252 11.3 Identities Associated with the Moduli 3. 6. and 12.........253 11.4 Identities Associated with the Modulus 7 11.5 False Theta Functions 12 Partial fractions..,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,..261 12.1 Introduction.,,,,,,,,,,,,,,,,,,,,,,,....261 12.2 The Basic Partial Fractions 12.3 Applications of the Partial Fraction Decompositions 12.4 Partial Fractions Plus 12.5 Related Identities ......................................279 12.6 Remarks on the Partial Fraction Method 13.2 Stieltjes-Wigert Polynomial............285 13 Hadamard Products for Two q-Series 13.1 Introduction 13.3 The Hadamard Factorization..............288 13. 4 Some Theta series 13.5 a Formal Power Series..,,,,,,,,,,,,,,,...,291 136 The Zeros of K。(2x) 13.7 Small Zeros of Koo(z) 13.8 A New Polynomial Sequence 13.9 The Zeros of pn(a) 13.10 A Theta Function Expansion 13.11 Ramanujan's Product for poo(a) 14 Integrals of Theta Functions 1 Introduction 309 14.2 Preliminary Results 14.3 The Identities on Page 207 14.4 Integral Representations of the Rogers-Ramanujan Continued fraction 15 Incomplete Elliptic Integrals 15.1 Introduction....................,.327 15.2 Preliminary Result 15.3 Two Simpler Integrals ...............:..................33015.4 Elliptic Integrals of Order 51) 15.5 Elliptic Integrals of Order 5(II) 15.6 Elliptic Integrals of Order 5(III)............34215.7 Elliptic Integrals of Order 15.............................349 15.8 Elliptic Integrals of Order 14 15.9 An Elliptic Integral of Order 35 15.10 Constructions of New Incomplete Elliptic Integral Identities.365 16 Infinite Integrals of q-Products 16.1 Introduction 16.2 Proofs 17 Modular Equations in Ramanujan's Lost Notebook....37317..13IntroductionSummaryof Modular Equations of Six Kind 172Eta- Function Identities.…………………375 4 A Fragment on Page 349 18 Fragments on Lambert Series...............395 18.1 Introduction...................... 395 18.2 Entries from the Two Fragments.............396 Location guide Provenance.... References Index..,,,,,,,,, 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.438 附录II刘治国教授访谈 编辑手记