Ramanujan's Lost Notebook

Part I

By: George E. Andrews, Bruce C. Berndt

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Hardcover | 6 May 2005

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452 Pages

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23.5 x 15.88 x 3.18

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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." Its discovery has frequently been deemed the mathematical equivalent of finding Beethoven's tenth symphony.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. Mathematicians are probably several decades away from a complete understanding of those functions. More than half of the material in the book is on q-series, including mock theta functions; the remaining part deals with theta function identities, modular equations, incomplete elliptic integrals of the first kind and other integrals of theta functions, Eisenstein series, particular values of theta functions, the Rogers-Ramanujan continued fraction, other q-continued fractions, other integrals, and parts of Hecke's theory of modular forms.

Industry Reviews

From the reviews: "The 'lost notebook' was in fact a 138-page manuscript found in materials from the estate of G.N. Watson. The manuscript, written in 'Ramanujan's distinctive handwriting', contained over 600 formulas. The authors have taken these results, provided proofs, placed them in the context of contemporary mathematics, and organized them accordingly ... This book is for the true fans of ... Ramanujan (Ramanuphiles?). If you enjoyed the original Ramanujan's Notebook series, then it's hard to pass this up." (Donald L. Vestal, MathDL-online, October 2006) "The present work is the first of an estimated four volumes devoted to all of the claims made by Ramanujan ... . The mathematics community owes a huge debt of gratitude to Andrews and Berndt for undertaking the monumental task of producing a coherent presentation along with complete proofs of the ... mathematical thoughts of Ramanujan during the last year of his life. ... Practitioners of q-series and other mathematicians interested in the work of Ramanujan, will delight in studying this book ... ." (Andrew V. Sills, Mathematical Reviews, Issue 2005 m)

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Preface	p. ix
Introduction	p. 1
The Rogers-Ramanujan Continued Fraction and Its Modular Properties	p. 9
Introduction	p. 9
Two-Variable Generalizations of (1.1.10) and (1.1.11)	p. 13
Hybrids of (1.1.10) and (1.1.11)	p. 18
Factorizations of (1.1.10) and (1.1.11)	p. 21
Modular Equations	p. 24
Theta-Function Identities of Degree 5	p. 26
Refinements of the Previous Identities	p. 28
Identities Involving the Parameter k = R(q)R[superscript 2](q[superscript 2])	p. 33
Other Representations of Theta Functions Involving R(q)	p. 39
Explicit Formulas Arising from (1.1.11)	p. 44
Explicit Evaluations of the Rogers-Ramanujan Continued Fraction	p. 57
Introduction	p. 57
Explicit Evaluations Using Eta-Function Identities	p. 59
General Formulas for Evaluating R [characters not reproducible] and S [characters not reproducible]	p. 66
Page 210 of Ramanujan's Lost Notebook	p. 71
Some Theta-Function Identities	p. 75
Ramanujan's General Explicit Formulas for the Rogers-Ramanujan Continued Fraction	p. 79
A Fragment on the Rogers-Ramanujan and Cubic Continued Fractions	p. 85
Introduction	p. 85
The Rogers-Ramanujan Continued Fraction	p. 86
The Theory of Ramanujan's Cubic Continued Fraction	p. 94
Explicit Evaluations of G(q)	p. 100
The Rogers-Ramanujan Continued Fraction and Its Partitions and Lambert Series	p. 107
Introduction	p. 107
Connections with Partitions	p. 108
Further Identities Involving the Power Series Coefficients of C(q) and 1/C(q)	p. 114
Generalized Lambert Series	p. 116
Further q-Series Representations for C(q)	p. 121
Finite Rogers-Ramanujan Continued Fractions	p. 125
Introduction	p. 125
Finite Rogers-Ramanujan Continued Fractions	p. 126
A generalization of Entry 5.2.1	p. 133
Class Invariants	p. 137
A Finite Generalized Rogers-Ramanujan Continued Fraction	p. 140
Other q-continued Fractions	p. 143
Introduction	p. 143
The Main Theorem	p. 144
A Second General Continued Fraction	p. 158
A Third General Continued Fraction	p. 159
A Transformation Formula	p. 162
Zeros	p. 165
Two Entries on Page 200 of Ramanujan's Lost Notebook	p. 169
An Elementary Continued Fraction	p. 172
Asymptotic Formulas for Continued Fractions	p. 179
Introduction	p. 179
The Main Theorem	p. 181
Two Asymptotic Formulas Found on Page 45 of Ramanujan's Lost Notebook	p. 187
An Asymptotic Formula for R(a,q)	p. 193
Ramanujan's Continued Fraction for (q[superscript 2];q[superscript 3])[infinity]/(q;q[superscript 3])[infinity]	p. 197
Introduction	p. 197
A Proof of Ramanujan's Formula (8.1.2)	p. 199
The Special Case a = w of (8.1.2)	p. 210
Two Continued Fractions Related to (q[superscript 2];q[superscript 3])[infinity]/(q;q[superscript 3])[infinity]	p. 213
An Asymptotic Expansion	p. 214
The Rogers-Fine Identity	p. 223
Introduction	p. 223
Series Transformations	p. 223
The Series [characters not reproducible]	p. 227
The Series [characters not reproducible]	p. 232
The Series [characters not reproducible]	p. 237
An Empirical Study of the Rogers-Ramanujan Identities	p. 241
Introduction	p. 241
The First Argument	p. 241
The Second Argument	p. 247
The Third Argument	p. 247
The Fourth Argument	p. 248
Rogers-Ramanujan-Slater-Type Identities	p. 251
Introduction	p. 251
Identities Associated with Modulus 5	p. 252
Identities Associated with the Moduli 3, 6, and 12	p. 253
Identities Associated with the Modulus 7	p. 256
False Theta Functions	p. 256
Partial Fractions	p. 261
Introduction	p. 261
The Basic Partial Fractions	p. 262
Applications of the Partial Fraction Decompositions	p. 265
Partial Fractions Plus	p. 272
Related Identities	p. 279
Remarks on the Partial Fraction Method	p. 284
Hadamard Products for Two q-Series	p. 285
Introduction	p. 285
Stieltjes-Wigert Polynomials	p. 286
The Hadamard Factorization	p. 288
Some Theta Series	p. 289
A Formal Power Series	p. 291
The Zeros of K[subscript infinity](zx)	p. 295
Small Zeros of K[subscript infinity](z)	p. 297
A New Polynomial Sequence	p. 297
The Zeros of p[subscript n](a)	p. 302
A Theta Function Expansion	p. 304
Ramanujan's Product for p[subscript infinity](a)	p. 305
Integrals of Theta Functions	p. 309
Introduction	p. 309
Preliminary Results	p. 310
The Identities on Page 207	p. 314
Integral Representations of the Rogers-Ramanujan Continued Fraction	p. 323
Incomplete Elliptic Integrals	p. 327
Introduction	p. 327
Preliminary Results	p. 328
Two Simpler Integrals	p. 330
Elliptic Integrals of Order 5 (I)	p. 333
Elliptic Integrals of Order 5 (II)	p. 339
Elliptic Integrals of Order 5 (III)	p. 342
Elliptic Integrals of Order 15	p. 349
Elliptic Integrals of Order 14	p. 356
An Elliptic Integral of Order 35	p. 361
Constructions of New Incomplete Elliptic Integral Identities	p. 365
Infinite Integrals of q-Products	p. 367
Introduction	p. 367
Proofs	p. 368
Modular Equations in Ramanujan's Lost Notebook	p. 373
Introduction	p. 373
Eta-Function Identities	p. 375
Summary of Modular Equations of Six Kinds	p. 384
A Fragment on Page 349	p. 392
Fragments on Lambert Series	p. 395
Introduction	p. 395
Entries from the Two Fragments	p. 396
Location Guide	p. 409
Provenance	p. 415
References	p. 419
Index	p. 433
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Ramanujan's Lost Notebook

Part I

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