Title:
Ramanujan's lost notebook
Personal Author:
Publication Information:
New York, NY : Springer, 2005
ISBN:
9780387255293
Subject Term:
Added Author:
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Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
---|---|---|---|---|---|
Searching... | 30000004604843 | QA29.R3 A52 2005 | Open Access Book | Book | Searching... |
On Order
Summary
Summary
In the library at Trinity College, Cambridge in 1976, George Andrews of Pennsylvania State University discovered a sheaf of pages in the handwriting of Srinivasa Ramanujan. Soon designated as "Ramanujan's Lost Notebook," it contains considerable material on mock theta functions and undoubtedly dates from the last year of Ramanujan's life. In this book, the notebook is presented with additional material and expert commentary.
Table of Contents
Preface | p. ix |
Introduction | p. 1 |
1 The Rogers-Ramanujan Continued Fraction and Its Modular Properties | p. 9 |
1.1 Introduction | p. 9 |
1.2 Two-Variable Generalizations of (1.1.10) and (1.1.11) | p. 13 |
1.3 Hybrids of (1.1.10) and (1.1.11) | p. 18 |
1.4 Factorizations of (1.1.10) and (1.1.11) | p. 21 |
1.5 Modular Equations | p. 24 |
1.6 Theta-Function Identities of Degree 5 | p. 26 |
1.7 Refinements of the Previous Identities | p. 28 |
1.8 Identities Involving the Parameter k = R(q)R[superscript 2](q[superscript 2]) | p. 33 |
1.9 Other Representations of Theta Functions Involving R(q) | p. 39 |
1.10 Explicit Formulas Arising from (1.1.11) | p. 44 |
2 Explicit Evaluations of the Rogers-Ramanujan Continued Fraction | p. 57 |
2.1 Introduction | p. 57 |
2.2 Explicit Evaluations Using Eta-Function Identities | p. 59 |
2.3 General Formulas for Evaluating R [characters not reproducible] and S [characters not reproducible] | p. 66 |
2.4 Page 210 of Ramanujan's Lost Notebook | p. 71 |
2.5 Some Theta-Function Identities | p. 75 |
2.6 Ramanujan's General Explicit Formulas for the Rogers-Ramanujan Continued Fraction | p. 79 |
3 A Fragment on the Rogers-Ramanujan and Cubic Continued Fractions | p. 85 |
3.1 Introduction | p. 85 |
3.2 The Rogers-Ramanujan Continued Fraction | p. 86 |
3.3 The Theory of Ramanujan's Cubic Continued Fraction | p. 94 |
3.4 Explicit Evaluations of G(q) | p. 100 |
4 The Rogers-Ramanujan Continued Fraction and Its Partitions and Lambert Series | p. 107 |
4.1 Introduction | p. 107 |
4.2 Connections with Partitions | p. 108 |
4.3 Further Identities Involving the Power Series Coefficients of C(q) and 1/C(q) | p. 114 |
4.4 Generalized Lambert Series | p. 116 |
4.5 Further q-Series Representations for C(q) | p. 121 |
5 Finite Rogers-Ramanujan Continued Fractions | p. 125 |
5.1 Introduction | p. 125 |
5.2 Finite Rogers-Ramanujan Continued Fractions | p. 126 |
5.3 A generalization of Entry 5.2.1 | p. 133 |
5.4 Class Invariants | p. 137 |
5.5 A Finite Generalized Rogers-Ramanujan Continued Fraction | p. 140 |
6 Other q-continued Fractions | p. 143 |
6.1 Introduction | p. 143 |
6.2 The Main Theorem | p. 144 |
6.3 A Second General Continued Fraction | p. 158 |
6.4 A Third General Continued Fraction | p. 159 |
6.5 A Transformation Formula | p. 162 |
6.6 Zeros | p. 165 |
6.7 Two Entries on Page 200 of Ramanujan's Lost Notebook | p. 169 |
6.8 An Elementary Continued Fraction | p. 172 |
7 Asymptotic Formulas for Continued Fractions | p. 179 |
7.1 Introduction | p. 179 |
7.2 The Main Theorem | p. 181 |
7.3 Two Asymptotic Formulas Found on Page 45 of Ramanujan's Lost Notebook | p. 187 |
7.4 An Asymptotic Formula for R(a,q) | p. 193 |
8 Ramanujan's Continued Fraction for (q[superscript 2];q[superscript 3])[infinity]/(q;q[superscript 3])[infinity] | p. 197 |
8.1 Introduction | p. 197 |
8.2 A Proof of Ramanujan's Formula (8.1.2) | p. 199 |
8.3 The Special Case a = w of (8.1.2) | p. 210 |
8.4 Two Continued Fractions Related to (q[superscript 2];q[superscript 3])[infinity]/(q;q[superscript 3])[infinity] | p. 213 |
8.5 An Asymptotic Expansion | p. 214 |
9 The Rogers-Fine Identity | p. 223 |
9.1 Introduction | p. 223 |
9.2 Series Transformations | p. 223 |
9.3 The Series [characters not reproducible] | p. 227 |
9.4 The Series [characters not reproducible] | p. 232 |
9.5 The Series [characters not reproducible] | p. 237 |
10 An Empirical Study of the Rogers-Ramanujan Identities | p. 241 |
10.1 Introduction | p. 241 |
10.2 The First Argument | p. 241 |
10.3 The Second Argument | p. 247 |
10.4 The Third Argument | p. 247 |
10.5 The Fourth Argument | p. 248 |
11 Rogers-Ramanujan-Slater-Type Identities | p. 251 |
11.1 Introduction | p. 251 |
11.2 Identities Associated with Modulus 5 | p. 252 |
11.3 Identities Associated with the Moduli 3, 6, and 12 | p. 253 |
11.4 Identities Associated with the Modulus 7 | p. 256 |
11.5 False Theta Functions | p. 256 |
12 Partial Fractions | p. 261 |
12.1 Introduction | p. 261 |
12.2 The Basic Partial Fractions | p. 262 |
12.3 Applications of the Partial Fraction Decompositions | p. 265 |
12.4 Partial Fractions Plus | p. 272 |
12.5 Related Identities | p. 279 |
12.6 Remarks on the Partial Fraction Method | p. 284 |
13 Hadamard Products for Two q-Series | p. 285 |
13.1 Introduction | p. 285 |
13.2 Stieltjes-Wigert Polynomials | p. 286 |
13.3 The Hadamard Factorization | p. 288 |
13.4 Some Theta Series | p. 289 |
13.5 A Formal Power Series | p. 291 |
13.6 The Zeros of K[subscript infinity](zx) | p. 295 |
13.7 Small Zeros of K[subscript infinity](z) | p. 297 |
13.8 A New Polynomial Sequence | p. 297 |
13.9 The Zeros of p[subscript n](a) | p. 302 |
13.10 A Theta Function Expansion | p. 304 |
13.11 Ramanujan's Product for p[subscript infinity](a) | p. 305 |
14 Integrals of Theta Functions | p. 309 |
14.1 Introduction | p. 309 |
14.2 Preliminary Results | p. 310 |
14.3 The Identities on Page 207 | p. 314 |
14.4 Integral Representations of the Rogers-Ramanujan Continued Fraction | p. 323 |
15 Incomplete Elliptic Integrals | p. 327 |
15.1 Introduction | p. 327 |
15.2 Preliminary Results | p. 328 |
15.3 Two Simpler Integrals | p. 330 |
15.4 Elliptic Integrals of Order 5 (I) | p. 333 |
15.5 Elliptic Integrals of Order 5 (II) | p. 339 |
15.6 Elliptic Integrals of Order 5 (III) | p. 342 |
15.7 Elliptic Integrals of Order 15 | p. 349 |
15.8 Elliptic Integrals of Order 14 | p. 356 |
15.9 An Elliptic Integral of Order 35 | p. 361 |
15.10 Constructions of New Incomplete Elliptic Integral Identities | p. 365 |
16 Infinite Integrals of q-Products | p. 367 |
16.1 Introduction | p. 367 |
16.2 Proofs | p. 368 |
17 Modular Equations in Ramanujan's Lost Notebook | p. 373 |
17.1 Introduction | p. 373 |
17.2 Eta-Function Identities | p. 375 |
17.3 Summary of Modular Equations of Six Kinds | p. 384 |
17.4 A Fragment on Page 349 | p. 392 |
18 Fragments on Lambert Series | p. 395 |
18.1 Introduction | p. 395 |
18.2 Entries from the Two Fragments | p. 396 |
Location Guide | p. 409 |
Provenance | p. 415 |
References | p. 419 |
Index | p. 433 |