Ramanujan's Lost Notebook

Name: Ramanujan's Lost Notebook
Price: 71.4 USD
Availability: InStock
ISBN: 9780387255293

ISBN-10: 038725529X

ISBN-13: 9780387255293

Edition: 2005

Authors: George E. Andrews, Bruce C. Berndt

List price: $139.99

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Description:

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. …

Book details

List price: $139.99
Copyright year: 2005
Publisher: Springer New York
Publication date: 5/6/2005
Binding: Hardcover
Pages: 438
Size: 6.10" wide x 9.25" long x 1.25" tall
Weight: 1.672
Language: English



Preface


Introduction



The Rogers-Ramanujan Continued Fraction and Its Modular Properties



Introduction



Two-Variable Generalizations of (1.1.10) and (1.1.11)



Hybrids of (1.1.10) and (1.1.11)



Factorizations of (1.1.10) and (1.1.11)



Modular Equations



Theta-Function Identities of Degree 5



Refinements of the Previous Identities



Identities Involving the Parameter k = R(q)R[superscript 2](q[superscript 2])



Other Representations of Theta Functions Involving R(q)



Explicit Formulas Arising from (1.1.11)



Explicit Evaluations of the Rogers-Ramanujan Continued Fraction



Introduction



Explicit Evaluations Using Eta-Function Identities



General Formulas for Evaluating R [characters not reproducible] and S [characters not reproducible]



Page 210 of Ramanujan's Lost Notebook



Some Theta-Function Identities



Ramanujan's General Explicit Formulas for the Rogers-Ramanujan Continued Fraction



A Fragment on the Rogers-Ramanujan and Cubic Continued Fractions



Introduction



The Rogers-Ramanujan Continued Fraction



The Theory of Ramanujan's Cubic Continued Fraction



Explicit Evaluations of G(q)



The Rogers-Ramanujan Continued Fraction and Its Partitions and Lambert Series



Introduction



Connections with Partitions



Further Identities Involving the Power Series Coefficients of C(q) and 1/C(q)



Generalized Lambert Series



Further q-Series Representations for C(q)



Finite Rogers-Ramanujan Continued Fractions



Introduction



Finite Rogers-Ramanujan Continued Fractions



A generalization of Entry 5.2.1



Class Invariants



A Finite Generalized Rogers-Ramanujan Continued Fraction



Other q-continued Fractions



Introduction



The Main Theorem



A Second General Continued Fraction



A Third General Continued Fraction



A Transformation Formula



Zeros



Two Entries on Page 200 of Ramanujan's Lost Notebook



An Elementary Continued Fraction



Asymptotic Formulas for Continued Fractions



Introduction



The Main Theorem



Two Asymptotic Formulas Found on Page 45 of Ramanujan's Lost Notebook



An Asymptotic Formula for R(a,q)



Ramanujan's Continued Fraction for (q[superscript 2];q[superscript 3])[infinity]/(q;q[superscript 3])[infinity]



Introduction



A Proof of Ramanujan's Formula (8.1.2)



The Special Case a = w of (8.1.2)



Two Continued Fractions Related to (q[superscript 2];q[superscript 3])[infinity]/(q;q[superscript 3])[infinity]



An Asymptotic Expansion



The Rogers-Fine Identity



Introduction



Series Transformations



The Series [characters not reproducible]



The Series [characters not reproducible]



The Series [characters not reproducible]



An Empirical Study of the Rogers-Ramanujan Identities



Introduction



The First Argument



The Second Argument



The Third Argument



The Fourth Argument



Rogers-Ramanujan-Slater-Type Identities



Introduction



Identities Associated with Modulus 5



Identities Associated with the Moduli 3, 6, and 12



Identities Associated with the Modulus 7



False Theta Functions



Partial Fractions



Introduction



The Basic Partial Fractions



Applications of the Partial Fraction Decompositions



Partial Fractions Plus



Related Identities



Remarks on the Partial Fraction Method



Hadamard Products for Two q-Series



Introduction



Stieltjes-Wigert Polynomials



The Hadamard Factorization



Some Theta Series



A Formal Power Series



The Zeros of K[subscript infinity](zx)



Small Zeros of K[subscript infinity](z)



A New Polynomial Sequence



The Zeros of p[subscript n](a)



A Theta Function Expansion



Ramanujan's Product for p[subscript infinity](a)



Integrals of Theta Functions



Introduction



Preliminary Results



The Identities on Page 207



Integral Representations of the Rogers-Ramanujan Continued Fraction



Incomplete Elliptic Integrals



Introduction



Preliminary Results



Two Simpler Integrals



Elliptic Integrals of Order 5 (I)



Elliptic Integrals of Order 5 (II)



Elliptic Integrals of Order 5 (III)



Elliptic Integrals of Order 15



Elliptic Integrals of Order 14



An Elliptic Integral of Order 35



Constructions of New Incomplete Elliptic Integral Identities



Infinite Integrals of q-Products



Introduction



Proofs



Modular Equations in Ramanujan's Lost Notebook



Introduction



Eta-Function Identities



Summary of Modular Equations of Six Kinds



A Fragment on Page 349



Fragments on Lambert Series



Introduction



Entries from the Two Fragments


Location Guide


Provenance


References


Index