Ramanujan s Lost Notebook

  • Filename: ramanujan-s-lost-notebook.
  • ISBN: 9781461438106
  • Release Date: 2012-06-08
  • Number of pages: 436
  • Author: George E Andrews
  • Publisher: Springer Science & Business Media



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plan to write on Ramanujan’s lost notebook and other manuscripts and fragments found in The Lost Notebook and Other Unpublished Papers, published by Narosa in 1988.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

Ramanujan s Lost Notebook

  • Filename: ramanujan-s-lost-notebook.
  • ISBN: 9781461440819
  • Release Date: 2013-06-04
  • Number of pages: 439
  • Author: George E Andrews
  • Publisher: Springer Science & Business Media



​​​​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​

Ramanujan s Notebooks

  • Filename: ramanujan-s-notebooks.
  • ISBN: 9781461208792
  • Release Date: 2012-12-06
  • Number of pages: 451
  • Author: Bruce C. Berndt
  • Publisher: Springer Science & Business Media



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.

Ramanujan

  • Filename: ramanujan.
  • ISBN: 0821826247
  • Release Date: 2001
  • Number of pages: 347
  • Author: Bruce C. Berndt
  • Publisher: American Mathematical Soc.



This book contains essays on Ramanujan and his work that were written especially for this volume. It also includes important survey articles in areas influenced by Ramanujan's mathematics. Most of the articles in the book are nontechnical, but even those that are more technical contain substantial sections that will engage the general reader. The book opens with the only four existing photographs of Ramanujan, presenting historical accounts of them and information about other people in the photos. This section includes an account of a cryptic family history written by his younger brother, S. Lakshmi Narasimhan. Following are articles on Ramanujan's illness by R. A. Rankin, the British physician D. A. B. Young, and Nobel laureate S. Chandrasekhar. They present a study of his symptoms, a convincing diagnosis of the cause of his death, and a thorough exposition of Ramanujan's life as a patient in English sanitariums and nursing homes. Following this are biographies of S. Janaki (Mrs. Ramanujan) and S. Narayana Iyer, Chief Accountant of the Madras Port Trust Office, who first communicated Ramanujan's work to the Journal of the Indian Mathematical Society. The last half of the book begins with a section on ``Ramanujan's Manuscripts and Notebooks''. Included is an important article by G. E. Andrews on Ramanujan's lost notebook. The final two sections feature both nontechnical articles, such as Jonathan and Peter Borwein's ``Ramanujan and pi'', and more technical articles by Freeman Dyson, Atle Selberg, Richard Askey, and G. N. Watson. This volume complements the book Ramanujan: Letters and Commentary, Volume 9, in the AMS series, History of Mathematics. For more on Ramanujan, see these AMS publications, Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, Volume 136.H, and Collected Papers of Srinivasa Ramanujan, Volume 159.H, in the AMS Chelsea Publishing series.

The Mathematical Legacy of Srinivasa Ramanujan

  • Filename: the-mathematical-legacy-of-srinivasa-ramanujan.
  • ISBN: 9788132207696
  • Release Date: 2012-10-06
  • Number of pages: 188
  • Author: M. Ram Murty
  • Publisher: Springer Science & Business Media



Srinivasa Ramanujan was a mathematician brilliant beyond comparison who inspired many great mathematicians. There is extensive literature available on the work of Ramanujan. But what is missing in the literature is an analysis that would place his mathematics in context and interpret it in terms of modern developments. The 12 lectures by Hardy, delivered in 1936, served this purpose at the time they were given. This book presents Ramanujan’s essential mathematical contributions and gives an informal account of some of the major developments that emanated from his work in the 20th and 21st centuries. It contends that his work still has an impact on many different fields of mathematical research. This book examines some of these themes in the landscape of 21st-century mathematics. These essays, based on the lectures given by the authors focus on a subset of Ramanujan’s significant papers and show how these papers shaped the course of modern mathematics.

Q series from a Contemporary Perspective

  • Filename: q-series-from-a-contemporary-perspective.
  • ISBN: 9780821811504
  • Release Date: 2000
  • Number of pages: 432
  • Author: Mourad Ismail
  • Publisher: American Mathematical Soc.



This volume presents the proceedings of the Joint Summer Research Conference on $q$-series, combinatorics, and computer algebra held at Mount Holyoke College (South Hadley, MA). All of the papers were contributed by participants and offer original research on topics of current interest. Articles in the book reflect the diversity of areas that overlap with $q$-series, as well as the usefulness of $q$-series across the mathematical sciences. The conference was held in honor of Richard Askey on the occasion of his 65th birthday and the proceedings contain an article about Askey's contributions to special functions.

Special Functions 2000 Current Perspective and Future Directions

  • Filename: special-functions-2000-current-perspective-and-future-directions.
  • ISBN: 9789401008181
  • Release Date: 2012-12-06
  • Number of pages: 520
  • Author: Joaquin Bustoz
  • Publisher: Springer Science & Business Media



The Advanced Study Institute brought together researchers in the main areas of special functions and applications to present recent developments in the theory, review the accomplishments of past decades, and chart directions for future research. Some of the topics covered are orthogonal polynomials and special functions in one and several variables, asymptotic, continued fractions, applications to number theory, combinatorics and mathematical physics, integrable systems, harmonic analysis and quantum groups, Painlevé classification.

Collected Papers of Srinivasa Ramanujan

  • Filename: collected-papers-of-srinivasa-ramanujan.
  • ISBN: 9781107536517
  • Release Date: 2015-12-03
  • Number of pages: 392
  • Author: Srinivasa Ramanujan
  • Publisher: Cambridge University Press



Originally published in 1927, this book presents the collected papers of the renowned Indian mathematician Srinivasa Ramanujan (1887-1920), with editorial contributions from G. H. Hardy (1877-1947). Detailed notes are incorporated throughout and appendices are also included. This book will be of value to anyone with an interest in the works of Ramanujan and the history of mathematics.

An Invitation to Q series

  • Filename: an-invitation-to-q-series.
  • ISBN: 9789814343855
  • Release Date: 2011
  • Number of pages: 236
  • Author: Chan Hei-Chi
  • Publisher: World Scientific



The aim of these lecture notes is to provide a self-contained exposition of several fascinating formulas discovered by Srinivasa Ramanujan. Two central results in these notes are: (1) the evaluation of the RogersOCoRamanujan continued fraction OCo a result that convinced G H Hardy that Ramanujan was a OC mathematician of the highest classOCO, and (2) what G. H. Hardy called Ramanujan''s OC Most Beautiful IdentityOCO. This book covers a range of related results, such as several proofs of the famous RogersOCoRamanujan identities and a detailed account of Ramanujan''s congruences. It also covers a range of techniques in q-series."

Number Theory in the Spirit of Ramanujan

  • Filename: number-theory-in-the-spirit-of-ramanujan.
  • ISBN: 9780821841785
  • Release Date: 2006
  • Number of pages: 187
  • Author: Bruce C. Berndt
  • Publisher: American Mathematical Soc.



Ramanujan is recognized as one of the great number theorists of the twentieth century. Here now is the first book to provide an introduction to his work in number theory. Most of Ramanujan's work in number theory arose out of $q$-series and theta functions. This book provides an introduction to these two important subjects and to some of the topics in number theory that are inextricably intertwined with them, including the theory of partitions, sums of squares and triangular numbers, and the Ramanujan tau function. The majority of the results discussed here are originally due to Ramanujan or were rediscovered by him. Ramanujan did not leave us proofs of the thousands of theorems he recorded in his notebooks, and so it cannot be claimed that many of the proofs given in this book are those found by Ramanujan. However, they are all in the spirit of his mathematics.The subjects examined in this book have a rich history dating back to Euler and Jacobi, and they continue to be focal points of contemporary mathematical research. Therefore, at the end of each of the seven chapters, Berndt discusses the results established in the chapter and places them in both historical and contemporary contexts. The book is suitable for advanced undergraduates and beginning graduate students interested in number theory.

Continued Fractions

  • Filename: continued-fractions.
  • ISBN: 9780821812006
  • Release Date: 1999
  • Number of pages: 379
  • Author: Leo Jerome Lange
  • Publisher: American Mathematical Soc.



This volume presents the contributions from the international conference held at the University of Missouri at Columbia, marking Professor Lange's 70th birthday and his retirement from the university. The principal purpose of the conference was to focus on continued fractions as a common interdisciplinary theme bridging gaps between a large number of fields--from pure mathematics to mathematical physics and approximation theory. Evident in this work is the widespread influence of continued fractions in a broad range of areas of mathematics and physics, including number theory, elliptic functions, Pade approximations, orthogonal polynomials, moment problems, frequency analysis, and regularity properties of evolution equations. Different areas of current research are represented. The lectures at the conference and the contributions to this volume reflect the wide range of applicability of continued fractions in mathematics and the applied sciences.

Number Theory and Modular Forms

  • Filename: number-theory-and-modular-forms.
  • ISBN: 9781475760446
  • Release Date: 2013-11-11
  • Number of pages: 400
  • Author: Bruce C. Berndt
  • Publisher: Springer Science & Business Media



Robert A. Rankin, one of the world's foremost authorities on modular forms and a founding editor of The Ramanujan Journal, died on January 27, 2001, at the age of 85. Rankin had broad interests and contributed fundamental papers in a wide variety of areas within number theory, geometry, analysis, and algebra. To commemorate Rankin's life and work, the editors have collected together 25 papers by several eminent mathematicians reflecting Rankin's extensive range of interests within number theory. Many of these papers reflect Rankin's primary focus in modular forms. It is the editors' fervent hope that mathematicians will be stimulated by these papers and gain a greater appreciation for Rankin's contributions to mathematics. This volume would be an inspiration to students and researchers in the areas of number theory and modular forms.

Ramanujan

  • Filename: ramanujan.
  • ISBN: 0821891251
  • Release Date: 1995-09-07
  • Number of pages: 347
  • Author: Srinivasa Ramanujan Aiyangar
  • Publisher: American Mathematical Soc.



The letters that Ramanujan wrote to G. H. Hardy on January 16 and February 27, 1913, are two of the most famous letters in the history of mathematics. These and other letters introduced Ramanujan and his remarkable theorems to the world and stimulated much research, especially in the 1920s and 1930s. This book brings together many letters to, from, and about Ramanujan. The letters came from the National Archives in Delhi, the Archives in the State of Tamil Nadu, and a variety of other sources. Helping to orient the reader is the extensive commentary, both mathematical and cultural, by Berndt and Rankin; in particular, they discuss in detail the history, up to the present day, of each mathematical result in the letters. Containing many letters that have never been published before, this book will appeal to those interested in Ramanujan's mathematics as well as those wanting to learn more about the personal side of his life. Ramanujan: Letters and Commentary was selected for the CHOICE list of Outstanding Academic Books for 1996.

A Synopsis of Elementary Results in Pure and Applied Mathematics Volume 1

  • Filename: a-synopsis-of-elementary-results-in-pure-and-applied-mathematics-volume-1.
  • ISBN: 9781108050678
  • Release Date: 2013-09-05
  • Number of pages: 298
  • Author: George Shoobridge Carr
  • Publisher: Cambridge University Press



When George Shoobridge Carr (1837-1914) wrote his Synopsis of Elementary Results he intended it as an aid to students preparing for degree-level examinations such as the Cambridge Mathematical Tripos, for which he provided private tuition. He would have been startled to see the two volumes, first published in 1880 and 1886 respectively, reissued more than a century later. Notably, in 1903 the work fell into the hands of the Indian prodigy Srinivasa Ramanujan (1887-1920) and greatly influenced his mathematical education. It is the interaction between a methodical teaching aid and the soaring spirit of a self-taught genius which gives this reissue its interest. Volume 1, presented here in its 1886 printing, contains sections on mathematical tables, algebra, the theory of equations, plane trigonometry, spherical trigonometry, elementary geometry and geometrical conics.

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