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Ramanujan's Lost Notebook Part II

:	George E Andrews, Bruce C. Berndt
:	Ramanujan's Lost Notebook Part II
:	Springer-Verlag
:	9780387777665
:	1
:	CHF 85.40
:

:	Arithmetik, Algebra
:	English

:	420
:	Wasserzeichen/DRM
:	PC/MAC/eReader/Tablet
:	PDF

In the spring of 1976, George Andrews of Pennsylvania State University visited the library at Trinity College, Cambridge, to examine the papers of the late G.N. Watson. Among these papers, Andrews discovered a sheaf of 138 pages in the handwriting of Srinivasa Ramanujan. This manuscript was soon designated 'Ramanujan's lost notebook.' The 'lost notebook' contains considerable material on mock theta functions and so undoubtedly emanates from the last year of Ramanujan's life. It should be emphasized that the material on mock theta functions is perhaps Ramanujan's deepest work.

	Preface	7
	Contents	8
	Introduction	12
	The Heine Transformation	16
	Introduction	16
	Heine's Method	17
	Ramanujan's Proof of the q-Gauss Summation Theorem	21
	Corollaries of (1.2.1) and (1.2.5)	25
	Corollaries of (1.2.6) and (1.2.7)	33
	Corollaries of (1.2.8), (1.2.9), and (1.2.10)	35
	Corollaries of Section 1.2 and Auxiliary Results	38
	The Sears--Thomae Transformation	55
	Introduction	55
	Direct Corollaries of (2.1.1) and (2.1.3)	55
	Extended Corollaries of (2.1.1) and (2.1.3)	56
	Bilateral Series	62
	Introduction	62
	Background	63
	The 11 Identity	65
	The 22 Identities	71
	Identities Arising from the Quintuple Product Identity	77
	Miscellaneous Bilateral Identities	83
	Well-Poised Series	89
	Introduction	89
	Applications of (4.1.3)	90
	Applications of Bailey's Formulas	97
	Bailey's Lemma and Theta Expansions	105
	Introduction	105
	The Main Lemma	105
	Corollaries of (5.2.3)	107
	Corollaries of (5.2.4) and Related Results	115
	Partial Theta Functions	121
	Introduction	121
	A General Identity	122
	Consequences of Theorem 6.2.1	123
	The function (a,q)	137
	Euler's Identity and Its Extensions	141
	The Warnaar Theory	149
	Special Identities	156
	Introduction	156
	Generalized Modular Relations	157
	Extending Abel's Lemma	165
	Innocents Abroad	172
	Theta Function Identities	180
	Introduction	180
	Cubic Identities	182
	Septic Identities	187
	Ramanujan's Cubic Class Invariant	202
	Introduction	202
	n and the Modular j-Invariant	206
	n and the Class Invariant Gn	210
	n and Modular Equations	211
	n and Modular Equations in the Theory of Signature 3	215
	n and Kronecker's Limit Formula	221
	The Remaining Five Values	224
	Some Modular Functions of Level 72	225
	Computations of n Using the Shimura Reciprocity Law	228
	Miscellaneous Results on Elliptic Functions and Theta Functions	232
	A Quasi-theta Product	232
	An Equivalent Formulation of (10.1.1) in Terms of Hyperbolic Series	233
	Further Remarks on Ramanujan's Quasi-theta Product	238
	A Generalization of the Dedekind Eta Function	241
	Two Entries on Page 346	245
	A Continued Fraction	247
	Class Invariants	248
	Formulas for the Power Series Coefficients of Certain Quotients of Eisenstein Series	250
	Introduction	250
	The Key Theorem	254
	The Coefficients of 1/Q(q)	264
	The Coefficients of Q(q)/R(q)	280
	The Coefficients of (P(q)/3)/R(q) and (P(q)/3)2/R(q)	287
	The Coefficients of (P(q)/23)/Q(q)	291
	Eight Identities for Eisenstein Series and Theta Functions	294
	The Coefficients of 1/B(q)	297
	Formulas for the Coefficients of Further Eisenstein Series	305
	The Coefficients of 1/B2(q)	307
	A Calculation from p	319
	Letters from Matlock House	320
	Introduction	320
	A Lower Bound	321
	An Upper Bound	329
	Eisenstein Series and Modular Equations	333
	Introduction	333
	Preliminary Results	334
	Quintic Identities: First Method	337
	Quintic Identities: Second Method	344
	Septic Identities	351
	Septic Differential Equations	359
	Series Representable in Terms of Eisenstein Series	361
	Introduction	361
	The Series T2k(q)	362
	The Series Un(q)	368
	Eisenstein Series and Approximations to	371
	Introduction	371
	Eisenstein Series and the Modular j-Invariant	372
	Eisenstein Series and Equations in : First Method	373
	Eisenstein Series and Equations in : Second Method	376
	Page 213	381
	Ramanujan's Series for 1/	381
	Miscellaneous Results on Eisenstein Series	391
	A generalization of Eisenstein Series	391
	Representations of Eisenstein Series in Terms of Elliptic Function Parameters	392
	Values of Certain Eisenstein Series	393
	Some Elementary Identities	394
	Location Guide	397
	Provenance	403
	References	406
	Index	420