国外很好数学著作原版系列拉马努金遗失笔记(第2卷) 内容简介

拉马努金数学遗失笔记,包括了S. Ramanujan在1988年由Narosa出版的《Lost Notebook and Other Unpublished Papers》和其他未发表的论文中提出的所有主张。这本书包含了“遗失的笔记”，它是1976年春天由作者在剑桥三一学院图书馆发现的。其中还包含其他部分手稿、碎片和拉马努金1917年-1919年在疗养院写给G.H.哈迪的信件。这是五卷中的第二卷，包含16章，包括314个条目。

国外很好数学著作原版系列拉马努金遗失笔记(第2卷) 目录

Contents Introduction.,,,,,,,,,,,,,,,,,,,,,,,,,,,,.,.,.1 1 The heine Transformation Introduction 1.2 Heine,'s Method 1.3 Ramanujan's Proof of the g-Gauss Summation Theorem... 10 4 Corollaries of (1.2. 1) and(1.2.5) 1.5 Corollaries of(1.2.)and(1.2.7) 1.6 Corollaries of(1.28),(1.2.9),and(1.2.10) 1.7 Corollaries of Section 1.2 and Auxiliary Results 2 The Sears-Thomae Transformation 2.1 Introduction 2.2 Direct Corollaries of (2.1.1)and(2.1.3) 2.3 Extended Corollaries of (2.1. 1)and(2.1.3) 3 Bilateral series 3.1 Introduction 3.2 Background...… 33 The 1/1 Identity..…… 3.4 The 20/2 Identities 3.5 Identities Arising from the Quintuple Product Identity ... 68 3.6 Miscellaneous bilateral Identities 4 Well-Poised series 4.1 Introduction 4.2 Applications of(4.1. 3) 4.3 Applications of Bailey' s Formulas........ 5 Bailey's Lemma and Theta Expansions 5.1 Introduction...… 5.2 The Main Lemma 5.3 Corollaries of (5.2.3 5.4 Corollaries of (5.2.4)and Related Results.........107 6 Partial Theta Functions 6.1 Introduction....…………… 6.2 A General Identity..……… 6.3 Consequences of Theorem6.2.1.…………115 6.4 The function v(a, q) 6.5 Euler's Identity and Its Extensions 6.6 The Warnaar Theory 7 Special Identities..…………………149 7.1 Introduction 7.2 Generalized Modular Relations.…… 7.3 Extending Abel's Lemma........ 7.4 Innocents Abroad 8 Theta Function Identities…………173 8.1 Introduction 8.2 Cubic Identities 8.3 Septic Identities Ramanujan's Cubic Class Invariant1 Introduction...................... 195 9.2 An and the Modular j- Invariant..……………199 9.3 An and the Class Invariant G,,,……203 9.4 An and Modular Equations 9.5 An and Modular Equations in the Theory of Signature 3. .. 208 6 An and Kronecker's Limit Formula9.7 The Remaining Five Values..……………217 9.9 Computations of入 Using the Shimura Reciprocity Lap…….218 9.8 Some Modular Functions of Level 72 10 Miscellaneous Results on Elliptic Functions and Theta Functions....,,.,,..,,,,.,,, 10.1 A Quasi-theta Product……225 10.2 An Equivalent Formulation of(10. 1.1)in Terms of Hyperbolic Series… 10.3 Further Remarks on Ramanujans Quasi-theta Product...... 231 10.4 A Generalization of the Dedekind Eta Function....... 234 10.5 Two Entries on Page346,…,…238 10.6 A Continued Fraction..……… 10.7 Class Invariants 11 Formulas for the Power Series Coefficients of Certain Quotients of Eisenstein Series...... 11.1 Introduction 112 The Key Theorem.………………247 11.3 The Coefficients of 1/Q(q) 11.4 The Coefficients of Q(o/R(q) 11.5 The Coefficients of(T P(a)/3)/R(g) and(P(a)/3)/R(q).. 280 1.6 The Coefficients of(πP(q)/2√3)/Q(q)…………………284 11.7 Eight Identities for Eisenstein Series and Theta Functions.. 287 118 The Coefficients of1/B(q)………290 11.9 Formulas for the Coefficients of Further Eisenstein Series 1.10 The Coefficients of 1/B2(q)……… 11.11 A Calculation from [176 12 Letters from Matlock House...........:.... 313 12.1 Introduction 12.2 A Lower Bound 12.3 An Upper Bound 13 Eisenstein Series and Modular Equations 327 13.1 Introduction ........ 13.2 Preliminary Results 13.3 Quintic Identities: First Method 13.4 Quintic Identities: Second Method..………338 13.5 Septic Identities 13.6 Septic Differential Equations……………353 14 Series Representable in Terms of Eisenstein Series 355 14.1 Introduction 355 14.2 The Series T2k(g) 14.3 The Series Un(q).………… 15 Eisenstein Series and Approximations to丌.………365 15.1 Introduction............... 15.2 Eisenstein Series and the Modular j-Invariant 15.3 Eisenstein Series and Equations in T: First Method ..... 367 15.4 Eisenstein Series and Equations in T: Second Method....37015.5Page213. 15.6 Ramanujan,'s Series for 1/T 16 Miscellaneous Results on Eisenstein Series.........385 16.1 A generalization of Eisenstein Series.……… 16.2 Representations of Eisenstein Series in Terms of Elliptic Function parameters 16.3 Values of Certain Eisenstein Series 16.4 Some Elementary Identities 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 附录II刘治国教授访谈 编辑手记

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