Ramanujan's Lost Notebook : Part V.

By: Andrews, George EContributor(s): Berndt, Bruce CMaterial type: TextTextSeries: eBooks on DemandPublisher: Cham : Springer, 2018Copyright date: ©2018Description: 1 online resource (433 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9783319778341Subject(s): Ramanujan Aiyangar, Srinivasa,-1887-1920 | Mathematical analysis | Mathematicians-India-BiographyGenre/Form: Electronic books.Additional physical formats: Print version:: Ramanujan's Lost Notebook : Part VDDC classification: 510.92 LOC classification: QA1-939Online resources: Click here to view this ebook.
Contents:
Intro -- Preface -- Contents -- 1 Introduction -- 2 Third Order Mock Theta Functions: ElementaryIdentities -- 2.1 Introduction -- 2.2 Basic Theorems -- 2.3 The Third Order Identities -- 3 Fifth Order Mock Theta Functions: ElementaryIdentities -- 3.1 Introduction -- 3.2 Basic Theorems -- 3.3 Watson's Fifth Order Identities -- 3.4 Ramanujan's Fifth Order Identities -- 3.5 Related Identities and Partitions -- 4 Third Order Mock Theta Functions: Partial Fraction Expansions -- 4.1 Introduction -- 4.2 Proofs of Entries 4.1.1-4.1.3 -- 4.3 Specializations -- 4.4 Proof of Entry 4.1.4. Part 1 -- 4.5 Proof of Entry 4.1.4. Part 2, Identities for Theta Functions and Lambert Series -- 4.6 Proof of Entry 4.1.4. Part 3, Proof of Theorem 4.1.1 -- 4.7 Proof of Entry 4.1.4. Part 4, Proof of Theorem 4.1.2 -- 5 The Mock Theta Conjectures: Equivalence -- 5.1 Introduction -- 5.2 Fourteen Lemmas -- 5.3 The Relations Among Mi(q), 1i5 -- 5.4 Relations to Partitions -- 6 Fifth Order Mock Theta Functions: Proof of the Mock Theta Conjectures -- 6.1 Introduction -- 6.2 Hecke-Type Series for f0(q) and f1(q) -- 6.3 Theta Function Identities -- 6.4 Partial Fractions and Appell-Lerch Series -- 6.5 Proof of the Mock Theta Conjectures -- 7 Sixth Order Mock Theta Functions -- 7.1 Introduction -- 7.2 Theta Function Identities -- 7.3 Hecke-Type Series for the Sixth Order Mock ThetaFunctions -- 7.4 Entries for φ6(q) and ψ6(q) -- 7.5 Entries for the Remaining Functions -- 7.6 Two Further Identities -- 7.7 Further Work -- 8 Tenth Order Mock Theta Functions: Part I, The First Four Identities -- 8.1 Introduction -- 8.2 Bailey Pairs -- 8.3 Hecke-Type Series -- 8.4 The First Four Tenth Order Identities: EquivalentFormulations -- 8.5 Proofs of Entries 8.4.1-8.4.4 -- 9 Tenth Order Mock Theta Functions: Part II, Identities for φ10(q), ψ10(q) -- 9.1 Introduction -- 9.2 A Preliminary Lemma.
9.3 φ10(q) and ψ10(q) as Power Series Coefficients -- 9.4 The Lambert Series L(z) and M(z) -- 9.5 Five-Dissection and Reformulation of D(z) -- 9.6 Further Decomposition of D(z) -- 9.7 Central Identities for ψ10(q) and φ10(q) -- 10 Tenth Order Mock Theta Functions: Part III, Identities for χ10(q), X10(q) -- 10.1 Introduction -- 10.2 A Preliminary Lemma -- 10.3 X10(q) and S10(q) as Coefficients -- 10.4 The Appell-Lerch Series L1(z) and M1(z) -- 10.5 Five-Dissection and Reformulation of E(z) -- 10.6 Further Decomposition of E(z) -- 10.7 Central Identities for X10(q) and χ10(q) -- 11 Tenth Order Mock Theta Functions: Part IV -- 11.1 Introduction -- 11.2 Properties of j(z -- q) -- 11.3 Properties of m(x,q,z) -- 11.4 Relating the Tenth Order Mock Theta Functionsto m(x,q,z) -- 12 Transformation Formulas: 10th Order Mock Theta Functions -- 12.1 Introduction -- 12.2 Some Theta Function Identities -- 12.3 Proof of Entry 12.1.1 -- 12.4 Proof of Entry 12.1.2 -- 12.5 Commentary -- 13 Two Identities Involving a Mordell Integral and Appell-Lerch Sums -- 13.1 Introduction -- 13.2 Two Lemmas -- 13.3 Proof of Theorem 13.1.1 -- 13.4 Proof of Theorem 13.1.2 -- 14 Ramanujan's Last Letter to Hardy -- 14.1 Introduction -- 14.2 The Last Letter -- 14.3 Formulas for the Taylor Series Coefficients of f3(q) -- 15 Euler Products in Ramanujan's Lost Notebook -- 15.1 Introduction -- 15.2 Scattered Entries on Euler Products -- 15.3 The Approach of Zhi-Hong Sun and Kenneth Williams Through the Theory of Binary Quadratic Forms -- 15.4 A Partial Manuscript on Euler Products -- 16 Continued Fractions -- 16.1 Introduction -- 16.2 Finite and Infinite Rogers-Ramanujan ContinuedFractions -- 17 Recent Work on Mock Theta Functions -- 17.1 Introduction -- 17.2 Zwegers' Insights -- 17.3 The Coefficients of Mock Theta Functions -- 17.4 Quantum Modular Forms and Beyond.
17.5 Combinatorial Interpretations -- 17.6 q-Series -- 18 Commentary on and Corrections to the First FourVolumes -- 18.1 Part I -- 18.2 Part II -- 18.3 Part III -- 18.4 Part IV -- 19 The Continuing Mystery -- 19.1 Introduction -- 19.2 The Rank of a Partition -- 19.3 The Role of Lerch's Transcendant and Basic Bilateral Hypergeometric Series -- 19.4 The Mock Theta Conjectures -- 19.5 The Seventh Order Mock Theta Functions -- 19.6 The Tenth Order Mock Theta Functions -- 19.7 Innocents Abroad (Still) -- 19.8 Identities for the Rogers-Ramanujan Functions -- 19.9 Hardy and Ramanujan on Sums of Squares -- 19.10 Puzzling Approximations -- 19.11 A Word of Caution -- Location Guide -- Provenance -- References -- Index.
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Electronic Book UT Tyler Online
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QA1-939 (Browse shelf) https://ebookcentral.proquest.com/lib/uttyler/detail.action?docID=5510037 Available EBC5510037

Intro -- Preface -- Contents -- 1 Introduction -- 2 Third Order Mock Theta Functions: ElementaryIdentities -- 2.1 Introduction -- 2.2 Basic Theorems -- 2.3 The Third Order Identities -- 3 Fifth Order Mock Theta Functions: ElementaryIdentities -- 3.1 Introduction -- 3.2 Basic Theorems -- 3.3 Watson's Fifth Order Identities -- 3.4 Ramanujan's Fifth Order Identities -- 3.5 Related Identities and Partitions -- 4 Third Order Mock Theta Functions: Partial Fraction Expansions -- 4.1 Introduction -- 4.2 Proofs of Entries 4.1.1-4.1.3 -- 4.3 Specializations -- 4.4 Proof of Entry 4.1.4. Part 1 -- 4.5 Proof of Entry 4.1.4. Part 2, Identities for Theta Functions and Lambert Series -- 4.6 Proof of Entry 4.1.4. Part 3, Proof of Theorem 4.1.1 -- 4.7 Proof of Entry 4.1.4. Part 4, Proof of Theorem 4.1.2 -- 5 The Mock Theta Conjectures: Equivalence -- 5.1 Introduction -- 5.2 Fourteen Lemmas -- 5.3 The Relations Among Mi(q), 1i5 -- 5.4 Relations to Partitions -- 6 Fifth Order Mock Theta Functions: Proof of the Mock Theta Conjectures -- 6.1 Introduction -- 6.2 Hecke-Type Series for f0(q) and f1(q) -- 6.3 Theta Function Identities -- 6.4 Partial Fractions and Appell-Lerch Series -- 6.5 Proof of the Mock Theta Conjectures -- 7 Sixth Order Mock Theta Functions -- 7.1 Introduction -- 7.2 Theta Function Identities -- 7.3 Hecke-Type Series for the Sixth Order Mock ThetaFunctions -- 7.4 Entries for φ6(q) and ψ6(q) -- 7.5 Entries for the Remaining Functions -- 7.6 Two Further Identities -- 7.7 Further Work -- 8 Tenth Order Mock Theta Functions: Part I, The First Four Identities -- 8.1 Introduction -- 8.2 Bailey Pairs -- 8.3 Hecke-Type Series -- 8.4 The First Four Tenth Order Identities: EquivalentFormulations -- 8.5 Proofs of Entries 8.4.1-8.4.4 -- 9 Tenth Order Mock Theta Functions: Part II, Identities for φ10(q), ψ10(q) -- 9.1 Introduction -- 9.2 A Preliminary Lemma.

9.3 φ10(q) and ψ10(q) as Power Series Coefficients -- 9.4 The Lambert Series L(z) and M(z) -- 9.5 Five-Dissection and Reformulation of D(z) -- 9.6 Further Decomposition of D(z) -- 9.7 Central Identities for ψ10(q) and φ10(q) -- 10 Tenth Order Mock Theta Functions: Part III, Identities for χ10(q), X10(q) -- 10.1 Introduction -- 10.2 A Preliminary Lemma -- 10.3 X10(q) and S10(q) as Coefficients -- 10.4 The Appell-Lerch Series L1(z) and M1(z) -- 10.5 Five-Dissection and Reformulation of E(z) -- 10.6 Further Decomposition of E(z) -- 10.7 Central Identities for X10(q) and χ10(q) -- 11 Tenth Order Mock Theta Functions: Part IV -- 11.1 Introduction -- 11.2 Properties of j(z -- q) -- 11.3 Properties of m(x,q,z) -- 11.4 Relating the Tenth Order Mock Theta Functionsto m(x,q,z) -- 12 Transformation Formulas: 10th Order Mock Theta Functions -- 12.1 Introduction -- 12.2 Some Theta Function Identities -- 12.3 Proof of Entry 12.1.1 -- 12.4 Proof of Entry 12.1.2 -- 12.5 Commentary -- 13 Two Identities Involving a Mordell Integral and Appell-Lerch Sums -- 13.1 Introduction -- 13.2 Two Lemmas -- 13.3 Proof of Theorem 13.1.1 -- 13.4 Proof of Theorem 13.1.2 -- 14 Ramanujan's Last Letter to Hardy -- 14.1 Introduction -- 14.2 The Last Letter -- 14.3 Formulas for the Taylor Series Coefficients of f3(q) -- 15 Euler Products in Ramanujan's Lost Notebook -- 15.1 Introduction -- 15.2 Scattered Entries on Euler Products -- 15.3 The Approach of Zhi-Hong Sun and Kenneth Williams Through the Theory of Binary Quadratic Forms -- 15.4 A Partial Manuscript on Euler Products -- 16 Continued Fractions -- 16.1 Introduction -- 16.2 Finite and Infinite Rogers-Ramanujan ContinuedFractions -- 17 Recent Work on Mock Theta Functions -- 17.1 Introduction -- 17.2 Zwegers' Insights -- 17.3 The Coefficients of Mock Theta Functions -- 17.4 Quantum Modular Forms and Beyond.

17.5 Combinatorial Interpretations -- 17.6 q-Series -- 18 Commentary on and Corrections to the First FourVolumes -- 18.1 Part I -- 18.2 Part II -- 18.3 Part III -- 18.4 Part IV -- 19 The Continuing Mystery -- 19.1 Introduction -- 19.2 The Rank of a Partition -- 19.3 The Role of Lerch's Transcendant and Basic Bilateral Hypergeometric Series -- 19.4 The Mock Theta Conjectures -- 19.5 The Seventh Order Mock Theta Functions -- 19.6 The Tenth Order Mock Theta Functions -- 19.7 Innocents Abroad (Still) -- 19.8 Identities for the Rogers-Ramanujan Functions -- 19.9 Hardy and Ramanujan on Sums of Squares -- 19.10 Puzzling Approximations -- 19.11 A Word of Caution -- Location Guide -- Provenance -- References -- Index.

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