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Autor George E. Andrews |
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Título : Ramanujan’s Lost Notebook : Part I Tipo de documento: documento electrónico Autores: George E. Andrews ; SpringerLink (Online service) ; Bruce C. Berndt Editorial: New York, NY : Springer New York Fecha de publicación: 2005 Número de páginas: XIV, 438 p Il.: online resource ISBN/ISSN/DL: 978-0-387-28124-7 Idioma : Inglés (eng) Palabras clave: Mathematics Algebraic geometry Sequences (Mathematics) Special functions Geometry Sequences, Series, Summability Functions Clasificación: 51 Matemáticas Resumen: This volume is the first of approximately four volumes devoted to providing statements, proofs, and discussions of all the claims made by Srinivasa Ramanujan in his lost notebook and all his other manuscripts and letters published with the lost notebook. In addition to the lost notebook, this publication contains copies of unpublished manuscripts in the Oxford library, in particular, his famous unpublished manuscript on the partition and tau-functions; fragments of both published and unpublished papers; miscellaneous sheets; and Ramanujan's letters to G. H. Hardy, written from nursing homes during Ramanujan's final two years in England. This volume contains accounts of 442 entries (counting multiplicities) made by Ramanujan in the aforementioned publication. The present authors have organized these claims into eighteen chapters, containing anywhere from two entries in Chapter 13 to sixty-one entries in Chapter 17. Most of the results contained in Ramanujan's Lost Notebook fall under the purview of q-series. These include mock theta functions, theta functions, partial theta function expansions, false theta functions, identities connected with the Rogers-Fine identity, several results in the theory of partitions, Eisenstein series, modular equations, the Rogers-Ramanujan continued fraction, other q-continued fractions, asymptotic expansions of q-series and q-continued fractions, integrals of theta functions, integrals of q-products, and incomplete elliptic integrals. Other continued fractions, other integrals, infinite series identities, Dirichlet series, approximations, arithmetic functions, numerical calculations, diophantine equations, and elementary mathematics are some of the further topics examined by Ramanujan in his lost notebook Nota de contenido: Inroduction -- Rogers-Ramanujan Continued Fraction and Its Modular Properties -- Explicit Evaluations of the Rogers-Ramanujan Continued Fraction -- A Fragment on the Rogers-Ramanujan and Cubic Continued Fractions -- The Rogers-Ramanujan Continued Fraction and Its Connections with Partitions and Lambert Series -- Finite Rogers-Ramanujan Continued Fractions -- Other q-continued Fractions -- Asymptotic Formulas for Continued Fractions -- Ramanujan’s Continued Fraction for (q2;q3)?/(q;q3)? -- The Rogers-Fine Identity -- An Empirical Study of the Rogers-Ramanujan Identities -- Rogers-Ramanujan-Slater Type Identities -- Partial Fractions -- Hadamard Products for Two q-Series -- Integrals of Theta Functions -- Incomplete Elliptic Integrals -- Infinite Integrals of q-Products -- Modular Equations in Ramanujan’s Lost Notebook -- Fragments on Lambert Series En línea: http://dx.doi.org/10.1007/0-387-28124-X Link: https://biblioteca.cunef.edu/gestion/catalogo/index.php?lvl=notice_display&id=35136 Ramanujan’s Lost Notebook : Part I [documento electrónico] / George E. Andrews ; SpringerLink (Online service) ; Bruce C. Berndt . - New York, NY : Springer New York, 2005 . - XIV, 438 p : online resource.
ISBN : 978-0-387-28124-7
Idioma : Inglés (eng)
Palabras clave: Mathematics Algebraic geometry Sequences (Mathematics) Special functions Geometry Sequences, Series, Summability Functions Clasificación: 51 Matemáticas Resumen: This volume is the first of approximately four volumes devoted to providing statements, proofs, and discussions of all the claims made by Srinivasa Ramanujan in his lost notebook and all his other manuscripts and letters published with the lost notebook. In addition to the lost notebook, this publication contains copies of unpublished manuscripts in the Oxford library, in particular, his famous unpublished manuscript on the partition and tau-functions; fragments of both published and unpublished papers; miscellaneous sheets; and Ramanujan's letters to G. H. Hardy, written from nursing homes during Ramanujan's final two years in England. This volume contains accounts of 442 entries (counting multiplicities) made by Ramanujan in the aforementioned publication. The present authors have organized these claims into eighteen chapters, containing anywhere from two entries in Chapter 13 to sixty-one entries in Chapter 17. Most of the results contained in Ramanujan's Lost Notebook fall under the purview of q-series. These include mock theta functions, theta functions, partial theta function expansions, false theta functions, identities connected with the Rogers-Fine identity, several results in the theory of partitions, Eisenstein series, modular equations, the Rogers-Ramanujan continued fraction, other q-continued fractions, asymptotic expansions of q-series and q-continued fractions, integrals of theta functions, integrals of q-products, and incomplete elliptic integrals. Other continued fractions, other integrals, infinite series identities, Dirichlet series, approximations, arithmetic functions, numerical calculations, diophantine equations, and elementary mathematics are some of the further topics examined by Ramanujan in his lost notebook Nota de contenido: Inroduction -- Rogers-Ramanujan Continued Fraction and Its Modular Properties -- Explicit Evaluations of the Rogers-Ramanujan Continued Fraction -- A Fragment on the Rogers-Ramanujan and Cubic Continued Fractions -- The Rogers-Ramanujan Continued Fraction and Its Connections with Partitions and Lambert Series -- Finite Rogers-Ramanujan Continued Fractions -- Other q-continued Fractions -- Asymptotic Formulas for Continued Fractions -- Ramanujan’s Continued Fraction for (q2;q3)?/(q;q3)? -- The Rogers-Fine Identity -- An Empirical Study of the Rogers-Ramanujan Identities -- Rogers-Ramanujan-Slater Type Identities -- Partial Fractions -- Hadamard Products for Two q-Series -- Integrals of Theta Functions -- Incomplete Elliptic Integrals -- Infinite Integrals of q-Products -- Modular Equations in Ramanujan’s Lost Notebook -- Fragments on Lambert Series En línea: http://dx.doi.org/10.1007/0-387-28124-X Link: https://biblioteca.cunef.edu/gestion/catalogo/index.php?lvl=notice_display&id=35136 Ejemplares
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Título : Ramanujan's Lost Notebook : Part II Tipo de documento: documento electrónico Autores: George E. Andrews ; SpringerLink (Online service) ; Bruce C. Berndt Editorial: New York, NY : Springer New York Fecha de publicación: 2009 Número de páginas: XII, 420 p. 8 illus Il.: online resource ISBN/ISSN/DL: 978-0-387-77766-5 Idioma : Inglés (eng) Palabras clave: Mathematics Algebraic geometry Sequences (Mathematics) Special functions Geometry Sequences, Series, Summability Functions Clasificación: 51 Matemáticas Resumen: This volume is the second of approximately four volumes that the authors plan to write on Ramanujan’s lost notebook, which is broadly interpreted to include all material published in The Lost Notebook and Other Unpublished Papers in 1988. The primary topics addressed in the authors’ second volume on the lost notebook are q-series, Eisenstein series, and theta functions. Most of the entries on q-series are located in the heart of the original lost notebook, while the entries on Eisenstein series are either scattered in the lost notebook or are found in letters that Ramanujan wrote to G.H. Hardy from nursing homes. About Ramanujan's Lost Notebook, Volume I: "Andrews and Berndt are to be congratulated on the job they are doing. This is the first step...on the way to an understanding of the work of the genius Ramanujan. It should act as an inspiration to future generations of mathematicians to tackle a job that will never be complete." - Gazette of the Australian Mathematical Society "...the results are organized topically with cross-references to the identities as they appear in the original Ramanujan manuscript. Particularly helpful are the extensive references, indicating where in the literature these results have been proven or independently discovered as well as where and how they have been used." - Bulletin of the American Mathematical Society "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 chaotically written mathematical thoughts of Ramanujan during the last year of his life. Some 85 years after his death, beautiful "new" and useful results of Ramanujan continue to be brought to light." - Mathematical Reviews Nota de contenido: The Heine Transformation -- The Sears#x2013; Thomae Transformation -- Bilateral Series -- Well-Poised Series -- Bailey#x02019;s Lemma and Theta Expansions -- Partial Theta Functions -- Special Identities -- Theta Function Identities -- Ramanujan#x02019;s Cubic Analogue of the Classical Ramanujan#x2013;Weber Class Invariants -- Miscellaneous Results on Elliptic Functions and Theta Functions -- Formulas for the Power Series Coefficients of Certain Quotients of Eisenstein Series -- Two Letters on Eisenstein Series Written from Matlock House -- Eisenstein Series and Modular Equations -- Series Representable in Terms of Eisenstein Series -- Eisenstein Series and Approximations to #x03C0; -- Miscellaneous Results on Eisenstein Series En línea: http://dx.doi.org/10.1007/b13290 Link: https://biblioteca.cunef.edu/gestion/catalogo/index.php?lvl=notice_display&id=33870 Ramanujan's Lost Notebook : Part II [documento electrónico] / George E. Andrews ; SpringerLink (Online service) ; Bruce C. Berndt . - New York, NY : Springer New York, 2009 . - XII, 420 p. 8 illus : online resource.
ISBN : 978-0-387-77766-5
Idioma : Inglés (eng)
Palabras clave: Mathematics Algebraic geometry Sequences (Mathematics) Special functions Geometry Sequences, Series, Summability Functions Clasificación: 51 Matemáticas Resumen: This volume is the second of approximately four volumes that the authors plan to write on Ramanujan’s lost notebook, which is broadly interpreted to include all material published in The Lost Notebook and Other Unpublished Papers in 1988. The primary topics addressed in the authors’ second volume on the lost notebook are q-series, Eisenstein series, and theta functions. Most of the entries on q-series are located in the heart of the original lost notebook, while the entries on Eisenstein series are either scattered in the lost notebook or are found in letters that Ramanujan wrote to G.H. Hardy from nursing homes. About Ramanujan's Lost Notebook, Volume I: "Andrews and Berndt are to be congratulated on the job they are doing. This is the first step...on the way to an understanding of the work of the genius Ramanujan. It should act as an inspiration to future generations of mathematicians to tackle a job that will never be complete." - Gazette of the Australian Mathematical Society "...the results are organized topically with cross-references to the identities as they appear in the original Ramanujan manuscript. Particularly helpful are the extensive references, indicating where in the literature these results have been proven or independently discovered as well as where and how they have been used." - Bulletin of the American Mathematical Society "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 chaotically written mathematical thoughts of Ramanujan during the last year of his life. Some 85 years after his death, beautiful "new" and useful results of Ramanujan continue to be brought to light." - Mathematical Reviews Nota de contenido: The Heine Transformation -- The Sears#x2013; Thomae Transformation -- Bilateral Series -- Well-Poised Series -- Bailey#x02019;s Lemma and Theta Expansions -- Partial Theta Functions -- Special Identities -- Theta Function Identities -- Ramanujan#x02019;s Cubic Analogue of the Classical Ramanujan#x2013;Weber Class Invariants -- Miscellaneous Results on Elliptic Functions and Theta Functions -- Formulas for the Power Series Coefficients of Certain Quotients of Eisenstein Series -- Two Letters on Eisenstein Series Written from Matlock House -- Eisenstein Series and Modular Equations -- Series Representable in Terms of Eisenstein Series -- Eisenstein Series and Approximations to #x03C0; -- Miscellaneous Results on Eisenstein Series En línea: http://dx.doi.org/10.1007/b13290 Link: https://biblioteca.cunef.edu/gestion/catalogo/index.php?lvl=notice_display&id=33870 Ejemplares
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Título : Ramanujan's Lost Notebook : Part III Tipo de documento: documento electrónico Autores: George E. Andrews ; SpringerLink (Online service) ; Bruce C. Berndt Editorial: New York, NY : Springer New York Fecha de publicación: 2012 Otro editor: Imprint: Springer Número de páginas: XII, 436 p Il.: online resource ISBN/ISSN/DL: 978-1-4614-3810-6 Idioma : Inglés (eng) Palabras clave: Mathematics Number theory Theory Clasificación: 51 Matemáticas Resumen: 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. This volume is the third of five volumes that the authors plan to write on Ramanujan’s lost notebook and other manuscripts and fragments found in The Lost Notebook and Other Unpublished Papers, published by Narosa in 1988. The ordinary partition function p(n) is the focus of this third volume. In particular, ranks, cranks, and congruences for p(n) are in the spotlight. Other topics include the Ramanujan tau-function, the Rogers–Ramanujan functions, highly composite numbers, and sums of powers of theta functions. Review from the second volume: "Fans of Ramanujan's mathematics are sure to be delighted by this book. While some of the content is taken directly from published papers, most chapters contain new material and some previously published proofs have been improved. Many entries are just begging for further study and will undoubtedly be inspiring research for decades to come. The next installment in this series is eagerly awaited." - MathSciNet Review from the first volume: "Andrews a nd Berndt are to be congratulated on the job they are doing. This is the first step...on the way to an understanding of the work of the genius Ramanujan. It should act as an inspiration to future generations of mathematicians to tackle a job that will never be complete." - Gazette of the Australian Mathematical Society Nota de contenido: Preface -- Introduction -- 1. Ranks and Cranks, Part I -- 2. Ranks and Cranks, Part II -- 3. Ranks and Cranks, Part III -- 4. Ramanujan's Unpublished Manuscript on the Partition and Tau Functions -- 5. Theorems about the Partition Function on Pages 189 and 182 -- 6. Congruences for Generalized Tau Functions on Page 178 -- 7. Ramanujan's Forty Identities for the Rogers-Ramanujan Functions -- 8. Circular Summation -- 9. Highly Composite Numbers -- Scratch Work -- Location Guide -- Provenance -- References En línea: http://dx.doi.org/10.1007/978-1-4614-3810-6 Link: https://biblioteca.cunef.edu/gestion/catalogo/index.php?lvl=notice_display&id=32821 Ramanujan's Lost Notebook : Part III [documento electrónico] / George E. Andrews ; SpringerLink (Online service) ; Bruce C. Berndt . - New York, NY : Springer New York : Imprint: Springer, 2012 . - XII, 436 p : online resource.
ISBN : 978-1-4614-3810-6
Idioma : Inglés (eng)
Palabras clave: Mathematics Number theory Theory Clasificación: 51 Matemáticas Resumen: 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. This volume is the third of five volumes that the authors plan to write on Ramanujan’s lost notebook and other manuscripts and fragments found in The Lost Notebook and Other Unpublished Papers, published by Narosa in 1988. The ordinary partition function p(n) is the focus of this third volume. In particular, ranks, cranks, and congruences for p(n) are in the spotlight. Other topics include the Ramanujan tau-function, the Rogers–Ramanujan functions, highly composite numbers, and sums of powers of theta functions. Review from the second volume: "Fans of Ramanujan's mathematics are sure to be delighted by this book. While some of the content is taken directly from published papers, most chapters contain new material and some previously published proofs have been improved. Many entries are just begging for further study and will undoubtedly be inspiring research for decades to come. The next installment in this series is eagerly awaited." - MathSciNet Review from the first volume: "Andrews a nd Berndt are to be congratulated on the job they are doing. This is the first step...on the way to an understanding of the work of the genius Ramanujan. It should act as an inspiration to future generations of mathematicians to tackle a job that will never be complete." - Gazette of the Australian Mathematical Society Nota de contenido: Preface -- Introduction -- 1. Ranks and Cranks, Part I -- 2. Ranks and Cranks, Part II -- 3. Ranks and Cranks, Part III -- 4. Ramanujan's Unpublished Manuscript on the Partition and Tau Functions -- 5. Theorems about the Partition Function on Pages 189 and 182 -- 6. Congruences for Generalized Tau Functions on Page 178 -- 7. Ramanujan's Forty Identities for the Rogers-Ramanujan Functions -- 8. Circular Summation -- 9. Highly Composite Numbers -- Scratch Work -- Location Guide -- Provenance -- References En línea: http://dx.doi.org/10.1007/978-1-4614-3810-6 Link: https://biblioteca.cunef.edu/gestion/catalogo/index.php?lvl=notice_display&id=32821 Ejemplares
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Título : Ramanujan's Lost Notebook : Part IV Tipo de documento: documento electrónico Autores: George E. Andrews ; SpringerLink (Online service) ; Bruce C. Berndt Editorial: New York, NY : Springer New York Fecha de publicación: 2013 Otro editor: Imprint: Springer Número de páginas: XVII, 439 p Il.: online resource ISBN/ISSN/DL: 978-1-4614-4081-9 Idioma : Inglés (eng) Palabras clave: Mathematics Mathematical analysis Analysis (Mathematics) Fourier Special functions Number theory Theory Functions Clasificación: 51 Matemáticas Resumen: 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. This volume is the fourth of five volumes that the authors plan to write on Ramanujan’s lost notebook. In contrast to the first three books on Ramanujan's Lost Notebook, the fourth book does not focus on q-series. Most of the entries examined in this volume fall under the purviews of number theory and classical analysis. Several incomplete manuscripts of Ramanujan published by Narosa with the lost notebook are discussed. Three of the partial manuscripts are on diophantine approximation, and others are in classical Fourier analysis and prime number theory. Most of the entries in number theory fall under the umbrella of classical analytic number theory. Perhaps the most intriguing entries are connected with the classical, unsolved circle and divisor problems. Review from the second volume: "Fans of Ramanujan's mathematics are sure to be delighted by this book. While some of the content is taken directly from published papers, most chapters contain new material and some previously published proofs have been improved. Many entries are just begging for further study and will undoubtedly be inspiring research for decades to come. The next installment in this series is eagerly awaited." - MathSciNet Review from the first volume: "Andrews and Berndt are to be congratulated on the job they are doing. This is the first step...on the way to an understanding of the work of the genius Ramanujan. It should act as an inspiration to future generations of mathematicians to tackle a job that will never be complete." - Gazette of the Australian Mathematical Society Nota de contenido: Preface -- 1 Introduction.- 2 Double Series of Bessel Functions and the Circle and Divisor Problems.- 3 Koshliakov's Formula and Guinand's Formula.- 4 Theorems Featuring the Gamma Function.- 5 Hypergeometric Series.- 6 Euler's Constant.- 7 Problems in Diophantine Approximation.- 8 Number Theory.- 9 Divisor Sums -- 10 Identities Related to the Riemann Zeta Function and Periodic Zeta Functions -- 11 Two Partial Unpublished Manuscripts on Sums Involving Primes.- 12 Infinite Series -- 13 A Partial Manuscript on Fourier and Laplace Transforms -- 14 Integral Analogues of Theta Functions adn Gauss Sums -- 15 Functional Equations for Products of Mellin Transforms -- 16 Infinite Products -- 17 A Preliminary Version of Ramanujan's Paper, On the Integral ?_0^x tan^(-1)t)/t dt -- 18 A Partial Manuscript Connected with Ramanujan's Paper, Some Definite Integrals.- 19 Miscellaneous Results in Analysis -- 20 Elementary Results -- 21 A Strange, Enigmatic Partial Manuscript.- Location Guide -- Provenance -- References -- Index En línea: http://dx.doi.org/10.1007/978-1-4614-4081-9 Link: https://biblioteca.cunef.edu/gestion/catalogo/index.php?lvl=notice_display&id=32213 Ramanujan's Lost Notebook : Part IV [documento electrónico] / George E. Andrews ; SpringerLink (Online service) ; Bruce C. Berndt . - New York, NY : Springer New York : Imprint: Springer, 2013 . - XVII, 439 p : online resource.
ISBN : 978-1-4614-4081-9
Idioma : Inglés (eng)
Palabras clave: Mathematics Mathematical analysis Analysis (Mathematics) Fourier Special functions Number theory Theory Functions Clasificación: 51 Matemáticas Resumen: 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. This volume is the fourth of five volumes that the authors plan to write on Ramanujan’s lost notebook. In contrast to the first three books on Ramanujan's Lost Notebook, the fourth book does not focus on q-series. Most of the entries examined in this volume fall under the purviews of number theory and classical analysis. Several incomplete manuscripts of Ramanujan published by Narosa with the lost notebook are discussed. Three of the partial manuscripts are on diophantine approximation, and others are in classical Fourier analysis and prime number theory. Most of the entries in number theory fall under the umbrella of classical analytic number theory. Perhaps the most intriguing entries are connected with the classical, unsolved circle and divisor problems. Review from the second volume: "Fans of Ramanujan's mathematics are sure to be delighted by this book. While some of the content is taken directly from published papers, most chapters contain new material and some previously published proofs have been improved. Many entries are just begging for further study and will undoubtedly be inspiring research for decades to come. The next installment in this series is eagerly awaited." - MathSciNet Review from the first volume: "Andrews and Berndt are to be congratulated on the job they are doing. This is the first step...on the way to an understanding of the work of the genius Ramanujan. It should act as an inspiration to future generations of mathematicians to tackle a job that will never be complete." - Gazette of the Australian Mathematical Society Nota de contenido: Preface -- 1 Introduction.- 2 Double Series of Bessel Functions and the Circle and Divisor Problems.- 3 Koshliakov's Formula and Guinand's Formula.- 4 Theorems Featuring the Gamma Function.- 5 Hypergeometric Series.- 6 Euler's Constant.- 7 Problems in Diophantine Approximation.- 8 Number Theory.- 9 Divisor Sums -- 10 Identities Related to the Riemann Zeta Function and Periodic Zeta Functions -- 11 Two Partial Unpublished Manuscripts on Sums Involving Primes.- 12 Infinite Series -- 13 A Partial Manuscript on Fourier and Laplace Transforms -- 14 Integral Analogues of Theta Functions adn Gauss Sums -- 15 Functional Equations for Products of Mellin Transforms -- 16 Infinite Products -- 17 A Preliminary Version of Ramanujan's Paper, On the Integral ?_0^x tan^(-1)t)/t dt -- 18 A Partial Manuscript Connected with Ramanujan's Paper, Some Definite Integrals.- 19 Miscellaneous Results in Analysis -- 20 Elementary Results -- 21 A Strange, Enigmatic Partial Manuscript.- Location Guide -- Provenance -- References -- Index En línea: http://dx.doi.org/10.1007/978-1-4614-4081-9 Link: https://biblioteca.cunef.edu/gestion/catalogo/index.php?lvl=notice_display&id=32213 Ejemplares
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