Ramanujan's notebooks

Author(s)

• Ramanujan Aiyangar, Srinivasa
• Berndt, Bruce C.

Bibliographic Information

Ramanujan's notebooks

Bruce C. Berndt

Springer-Verlag, c1985-1998

• pt. 1
• pt. 1 : softcover
• pt. 2
• pt. 2 : softcover
• pt. 3 : us
• pt. 3 : gw
• pt. 3 : softcover
• pt. 4 : us
• pt. 4 : gw
• pt. 4 : softcover
• pt. 5 : us
• pt. 5 : softcover

Note

Includes bibliographical references and index

Description and Table of Contents

Volume

pt. 4 : us ISBN 9780387941097

Description

During the years 1903-1914, Ramanujan worked in almost complete isolation in India. During this time, he recorded most of his mathematical discoveries without proofs in notebooks. Although many of his results were already found in the literature, most were not. Almost a decade after Ramanujan's death in 1920, G.N. Watson and B.M. Wilson began to edit Ramanujan's notebooks, but they never completed the task. A photostat edition, with no editing, was published by the Tata Institute of Fundamental Research in Bombay in 1957. This book is the fourth of five volumes devoted to the editing of Ramanujan's notebooks. Parts I, II, and III, published in 1985, 1989, and 1991, contain accounts of Chapters 1-21 in Ramanujan's second notebook as well as a description of his quarterly reports. This is the first of two volumes devoted to proving the results found in the unorganized portions of the second notebook and in the third notebook. The author also proves those results in the first notebook that are not found in the second or third notebooks. For those results that are known, references in the literature are provided. Otherwise, complete proofs are given. Over 1/2 of the results in the notebooks are new. Many of them are so startling and different that there are no results akin to them in the literature.

Table of Contents

22 Elementary Results.- 23 Number Theory.- 24 Ramanujan's Theory of Prime Numbers.- 25 Theta-Functions and Modular Equations.- 26 Inversion Formulas for the Lemniscate and Allied Functions.- 27 q-Series.- 28 Integrals.- 29 Special Functions.- 30 Partial Fraction Expansions.- 31 Elementary and Miscellaneous Analysis.- Location in Notebook 2 of the Material in the 16 Chapters of Notebook 1>.- References.

Volume

pt. 5 : us ISBN 9780387949413

Description

The fifth and final volume to establish the results claimed by the great Indian mathematician Srinivasa Ramanujan in his "Notebooks" first published in 1957. Although each of the five volumes contains many deep results, the average depth in this volume is possibly greater than in the first four. There are several results on continued fractions - a subject that Ramanujan loved very much. It is the authors wish that this and previous volumes will serve as springboards for further investigations by mathematicians intrigued by Ramanujans remarkable ideas.

Table of Contents

32 Continued Fractions.- 1 The Rogers-Ramanujan Continued Fraction.- 2 Other q-Continued Fractions.- 3 Continued Fractions Arising from Products of Gamma Functions.- 4 Other Continued Fractions.- 5 General Theorems.- 33 Ramanujan's Theories of Elliptic Functions to Alternative Bases.- 1 Introduction.- 2 Ramanujan's Cubic Transformation, the Borweins' Cubic Theta-Function Identity, and the Inversion Formula.- 3 The Principles of Triplication and Trimidiation.- 4 The Eisenstein Series L, M, and N.- 5 A Hypergeometric Transformation and Associated Transfer Principle.- 6 More Higher Order Transformations for Hypergeometric Series.- 7 Modular Equations in the Theory of Signature 3.- 8 The Inversion of an Analogue of K (k) in Signature 3.- 9 The Theory for Signature 4.- 10 Modular Equations in the Theory of Signature 4.- 11 The Theory for Signature 6.- 12 An Identity from the First Notebook and Further Hypergeometric Transformations.- 13 Some Enigmatic Formulas Near the End of the Third Notebook.- 14 Concluding Remarks.- 34 Class Invariants and Singular Moduli.- 1 Introduction.- 2 Table of Class Invariants.- 3 Computation of Gnand gnwhen 9/n.- 4 Kronecker's Limit Formula and General Formulas for Class Invariants.- 5 Class Invariants Via Kronecker's Limit Formula.- 6 Class Invariants Via Modular Equations.- 7 Class Invariants Via Class Field Theory.- 8 Miscellaneous Results.- 9 Singular Moduli.- 10 A Certain Rational Function of Singular Moduli.- 11 The Modular j-invariant.- 35 Values of Theta-Functions.- 0 Introduction.- 1 Elementary Values.- 2 Nonelementary Values of.- 3 A Remarkable Product of Theta-Functions.- 36 Modular Equations and Theta-Function Identities in Notebook 1.- 1 Modular Equations of Degree 3 and Related Theta-Function Identities.- 2 Modular Equations of Degree 5 and Related Theta-Function Identities.- 3 Other Modular Equations and Related Theta-Function Identities.- 4 Identities Involving Lambert Series.- 5 Identities Involving Eisenstein Series.- 6 Modular Equations in the Form of Schlafli.- 7 Modular Equations in the Form of Russell.- 8 Modular Equations in the Form of Weber.- 9 Series Transformations Associated with Theta-Functions.- 10 Miscellaneous Results.- 37 Infinite Series.- 38 Approximations and Asymptotic Expansions.- 39 Miscellaneous Results in the First Notebook.- Location of Entries in the Unorganized Portions of Ramanujan's First Notebook.- References.

Volume

pt. 1 ISBN 9780387961101

Description

Srinivasa Ramanujan is, arguably, the greatest mathematician that India has produced. His story is quite unusual: although he had no formal education inmathematics, he taught himself, and managed to produce many important new results. With the support of the English number theorist G. H. Hardy, Ramanujan received a scholarship to go to England and study mathematics. He died very young, at the age of 32, leaving behind three notebooks containing almost 3000 theorems, virtually all without proof. G. H. Hardy and others strongly urged that notebooks be edited and published, and the result is this series of books. This volume dealswith Chapters 1-9 of Book II; each theorem is either proved, or a reference to a proof is given.

Table of Contents

1 Magic Squares.- 2 Sums Related to the Harmonic Series or the Inverse Tangent function.- 3 Combinatorial Analysis and Series Inversions.- 4 Iterates of the Exponential Function and an Ingenious Formal Technique.- 5 Eulerian Polynomials and Numbers, Bernoulli Numbers, and the Riemann Zeta-Function.- 6 Ramanujan's Theory of Divergent Series.- 7 Sums of Powers, Bernoulli Numbers, and the Gamma function.- 8 Analogues of the Gamma function.- 9 Infinite Series Identities, Transformations, and Evaluations.- Ramanujan's Quarterly Reports.- References.

Volume

pt. 2 ISBN 9780387967943

Description

During the years 1903-1914, Ramanujan recorded many of his mathematical discoveries in notebooks without providing proofs. Although many of his results were already in the literature, more were not. Almost a decade after Ramanujan's death in 1920, G.N. Watson and B.M. Wilson began to edit his notebooks but never completed the task. A photostat edition, with no editing, was published by the Tata Institute of Fundamental Research in Bombay in 1957. This book is the second of four volumes devoted to the editing of Ramanujan's Notebooks. Part I, published in 1985, contains an account of Chapters 1-9 in the second notebook as well as a description of Ramanujan's quarterly reports. In this volume, we examine Chapters 10-15 in Ramanujan's second notebook. If a result is known, we provide references in the literature where proofs may be found; if a result is not known, we attempt to prove it. Not only are the results fascinating, but, for the most part, Ramanujan's methods remain a mystery. Much work still needs to be done. We hope readers will strive to discover Ramanujan's thoughts and further develop his beautiful ideas.

Table of Contents

10 Hypergeometric Series, I.- 11 Hypergeometric Series, II.- 12 Continued Fractions.- 13 Integrals and Asymptotic Expansions.- 14 Infinite Series.- 15 Asymptotic Expansions and Modular Forms.- References.

Volume

pt. 3 : us ISBN 9780387975030

Description

Upon Ramanujans death in 1920, G. H. Hardy strongly urged that Ramanujans notebooks be published and edited. In 1957, the Tata Institute of Fundamental Research in Bombay finally published a photostat edition of the notebooks, but no editing was undertaken. In 1977, Berndt began the task of editing Ramanujans notebooks: proofs are provided to theorems not yet proven in previous literature, and many results are so startling as to be unique.

Table of Contents

16 q-Series and Theta-Functions.- 17 Fundamental Properties of Elliptic Functions.- 18 The Jacobian Elliptic Functions.- 19 Modular Equations of Degrees 3, 5, and 7 and Associated Theta-Function Identities.- 20 Modular Equations of Higher and Composite Degrees.- 21 Eisenstein Series.- References.

Volume

pt. 4 : softcover ISBN 9781461269328

Description

During the years 1903-1914, Ramanujan worked in almost complete isolation in India. During this time, he recorded most of his mathematical discoveries without proofs in notebooks. Although many of his results were already found in the literature, most were not. Almost a decade after Ramanujan's death in 1920, G.N. Watson and B.M. Wilson began to edit Ramanujan's notebooks, but they never completed the task. A photostat edition, with no editing, was published by the Tata Institute of Fundamental Research in Bombay in 1957. This book is the fourth of five volumes devoted to the editing of Ramanujan's notebooks. Parts I, II, and III, published in 1985, 1989, and 1991, contain accounts of Chapters 1-21 in Ramanujan's second notebook as well as a description of his quarterly reports. This is the first of two volumes devoted to proving the results found in the unorganized portions of the second notebook and in the third notebook. The author also proves those results in the first notebook that are not found in the second or third notebooks. For those results that are known, references in the literature are provided. Otherwise, complete proofs are given. Over 1/2 of the results in the notebooks are new. Many of them are so startling and different that there are no results akin to them in the literature.

Table of Contents

22 Elementary Results.- 23 Number Theory.- 24 Ramanujan's Theory of Prime Numbers.- 25 Theta-Functions and Modular Equations.- 26 Inversion Formulas for the Lemniscate and Allied Functions.- 27 q-Series.- 28 Integrals.- 29 Special Functions.- 30 Partial Fraction Expansions.- 31 Elementary and Miscellaneous Analysis.- Location in Notebook 2 of the Material in the 16 Chapters of Notebook 1>.- References.

Volume

pt. 3 : softcover ISBN 9781461269632

Description

Upon Ramanujans death in 1920, G. H. Hardy strongly urged that Ramanujans notebooks be published and edited. In 1957, the Tata Institute of Fundamental Research in Bombay finally published a photostat edition of the notebooks, but no editing was undertaken. In 1977, Berndt began the task of editing Ramanujans notebooks: proofs are provided to theorems not yet proven in previous literature, and many results are so startling as to be unique.

Table of Contents

16 q-Series and Theta-Functions.- 17 Fundamental Properties of Elliptic Functions.- 18 The Jacobian Elliptic Functions.- 19 Modular Equations of Degrees 3, 5, and 7 and Associated Theta-Function Identities.- 20 Modular Equations of Higher and Composite Degrees.- 21 Eisenstein Series.- References.

Volume

pt. 1 : softcover ISBN 9781461270072

Description

Srinivasa Ramanujan is, arguably, the greatest mathematician that India has produced. His story is quite unusual: although he had no formal education inmathematics, he taught himself, and managed to produce many important new results. With the support of the English number theorist G. H. Hardy, Ramanujan received a scholarship to go to England and study mathematics. He died very young, at the age of 32, leaving behind three notebooks containing almost 3000 theorems, virtually all without proof. G. H. Hardy and others strongly urged that notebooks be edited and published, and the result is this series of books. This volume dealswith Chapters 1-9 of Book II; each theorem is either proved, or a reference to a proof is given.

Table of Contents

1 Magic Squares.- 2 Sums Related to the Harmonic Series or the Inverse Tangent function.- 3 Combinatorial Analysis and Series Inversions.- 4 Iterates of the Exponential Function and an Ingenious Formal Technique.- 5 Eulerian Polynomials and Numbers, Bernoulli Numbers, and the Riemann Zeta-Function.- 6 Ramanujan's Theory of Divergent Series.- 7 Sums of Powers, Bernoulli Numbers, and the Gamma function.- 8 Analogues of the Gamma function.- 9 Infinite Series Identities, Transformations, and Evaluations.- Ramanujan's Quarterly Reports.- References.

Volume

pt. 5 : softcover ISBN 9781461272212

Description

The fifth and final volume to establish the results claimed by the great Indian mathematician Srinivasa Ramanujan in his "Notebooks" first published in 1957. Although each of the five volumes contains many deep results, the average depth in this volume is possibly greater than in the first four. There are several results on continued fractions - a subject that Ramanujan loved very much. It is the authors wish that this and previous volumes will serve as springboards for further investigations by mathematicians intrigued by Ramanujans remarkable ideas.

Table of Contents

32 Continued Fractions.- 1 The Rogers-Ramanujan Continued Fraction.- 2 Other q-Continued Fractions.- 3 Continued Fractions Arising from Products of Gamma Functions.- 4 Other Continued Fractions.- 5 General Theorems.- 33 Ramanujan's Theories of Elliptic Functions to Alternative Bases.- 1 Introduction.- 2 Ramanujan's Cubic Transformation, the Borweins' Cubic Theta-Function Identity, and the Inversion Formula.- 3 The Principles of Triplication and Trimidiation.- 4 The Eisenstein Series L, M, and N.- 5 A Hypergeometric Transformation and Associated Transfer Principle.- 6 More Higher Order Transformations for Hypergeometric Series.- 7 Modular Equations in the Theory of Signature 3.- 8 The Inversion of an Analogue of K (k) in Signature 3.- 9 The Theory for Signature 4.- 10 Modular Equations in the Theory of Signature 4.- 11 The Theory for Signature 6.- 12 An Identity from the First Notebook and Further Hypergeometric Transformations.- 13 Some Enigmatic Formulas Near the End of the Third Notebook.- 14 Concluding Remarks.- 34 Class Invariants and Singular Moduli.- 1 Introduction.- 2 Table of Class Invariants.- 3 Computation of Gnand gnwhen 9/n.- 4 Kronecker's Limit Formula and General Formulas for Class Invariants.- 5 Class Invariants Via Kronecker's Limit Formula.- 6 Class Invariants Via Modular Equations.- 7 Class Invariants Via Class Field Theory.- 8 Miscellaneous Results.- 9 Singular Moduli.- 10 A Certain Rational Function of Singular Moduli.- 11 The Modular j-invariant.- 35 Values of Theta-Functions.- 0 Introduction.- 1 Elementary Values.- 2 Nonelementary Values of.- 3 A Remarkable Product of Theta-Functions.- 36 Modular Equations and Theta-Function Identities in Notebook 1.- 1 Modular Equations of Degree 3 and Related Theta-Function Identities.- 2 Modular Equations of Degree 5 and Related Theta-Function Identities.- 3 Other Modular Equations and Related Theta-Function Identities.- 4 Identities Involving Lambert Series.- 5 Identities Involving Eisenstein Series.- 6 Modular Equations in the Form of Schlafli.- 7 Modular Equations in the Form of Russell.- 8 Modular Equations in the Form of Weber.- 9 Series Transformations Associated with Theta-Functions.- 10 Miscellaneous Results.- 37 Infinite Series.- 38 Approximations and Asymptotic Expansions.- 39 Miscellaneous Results in the First Notebook.- Location of Entries in the Unorganized Portions of Ramanujan's First Notebook.- References.

Volume

pt. 2 : softcover ISBN 9781461288657

Description

During the years 1903-1914, Ramanujan recorded many of his mathematical discoveries in notebooks without providing proofs. Although many of his results were already in the literature, more were not. Almost a decade after Ramanujan's death in 1920, G.N. Watson and B.M. Wilson began to edit his notebooks but never completed the task. A photostat edition, with no editing, was published by the Tata Institute of Fundamental Research in Bombay in 1957. This book is the second of four volumes devoted to the editing of Ramanujan's Notebooks. Part I, published in 1985, contains an account of Chapters 1-9 in the second notebook as well as a description of Ramanujan's quarterly reports. In this volume, we examine Chapters 10-15 in Ramanujan's second notebook. If a result is known, we provide references in the literature where proofs may be found; if a result is not known, we attempt to prove it. Not only are the results fascinating, but, for the most part, Ramanujan's methods remain a mystery. Much work still needs to be done. We hope readers will strive to discover Ramanujan's thoughts and further develop his beautiful ideas.

Table of Contents

10 Hypergeometric Series, I.- 11 Hypergeometric Series, II.- 12 Continued Fractions.- 13 Integrals and Asymptotic Expansions.- 14 Infinite Series.- 15 Asymptotic Expansions and Modular Forms.- References.

Volume

pt. 4 : gw ISBN 9783540941095

Description

During the years 1903-1914, Ramanujan worked in almost complete isolation in India. During this time, he recorded most of his mathematical discoveries without proofs in notebooks. Although many of his results were already found in the literature, most were not. Almost a decade after Ramanujan's death in 1920, G.N. Watson and B.M. Wilson began to edit Ramanujan's notebooks, but they never completed the task. A photostat edition, with no editing, was published by the Tata Institute of Fundamental Research in Bombay in 1957. This book is the fourth of five volumes devoted to the editing of Ramanujan's notebooks. Parts I, II, and III, published in 1985, 1989, and 1991, contain accounts of Chapters 1-21 in Ramanujan's second notebook as well as a description of his quarterly reports. This is the first of two volumes devoted to proving the results found in the unorganized portions of the second notebook and in the third notebook. The author also proves those results in the first notebook that are not found in the second or third notebooks. For those results that are known, references in the literature are provided. Otherwise, complete proofs are given. Over 1/2 of the results in the notebooks are new. Many of them are so startling and different that there are no results akin to them in the literature.

Volume

pt. 3 : gw ISBN 9783540975038

Description

During the time period between 1903 and 1914, Ramanujan worked in almost complete isolation in India. Throughout these years, he recorded his mathematical results without proofs in notebooks. Upon Ramanujan's death in 1920, G.H. Hardy strongly urged that Ramanujan's notebooks be published and edited. The English mathematicians G.N. Watson and B.M. Wilson began this task in 1929, but although they devoted nearly ten years to the project, the work was never completed. In 1957, the Tata Institute of Fundamental Research in Bombay published a photostat edition of the notebooks, but no editing was undertaken. In 1977, Berndt began the task of editing Ramanujan's notebooks. Proofs are provided to theorems not yet proven in previous literature, and many results are so startling and different that there are no results akin to them in the literature. This monograph on history of mathematics and number theory is intended for graduate students in mathematics.

by "Nielsen BookData"

Details

• NCID: BA00478837
• ISBN:
  • 0387961100
  • 9781461270072
  • 038796794X
  • 9781461288657
  • 0387975039
  • 3540975039
  • 9781461269632
  • 0387941096
  • 3540941096
  • 9781461269328
  • 0387949410
  • 9781461272212
• LCCN: 84020201
• Country Code: us
• Title Language Code: eng
• Text Language Code: eng
• Place of Publication: New York ; Tokyo
• Pages/Volumes: 5 v.
• Size: 25 cm
• Classification:
  • LCC : QA3
  • DC19 : 510
  • NDC8 : 410.8
• Subject Headings:
  • LCSH : Mathematics -- Collected works
  • NDLSH : 数学