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.

Author: George E. Andrews

Publisher: Springer

ISBN: 9783319778341

Category: Mathematics

Page: 430

View: 517

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 fifth and final installment of the authors’ examination of Ramanujan’s lost notebook focuses on the mock theta functions first introduced in Ramanujan’s famous Last Letter. This volume proves all of the assertions about mock theta functions in the lost notebook and in the Last Letter, particularly the celebrated mock theta conjectures. Other topics feature Ramanujan’s many elegant Euler products and the remaining entries on continued fractions not discussed in the preceding volumes. 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

S. Ramanujan's last letter to G. H. Hardy contains the definition of mock theta functions, a list of 17 mock theta functions, and 5 identities related to mock theta functions.

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 ...

Author: Srinivasa Ramanujan AiyangarPublish On: 1988

The so-called Lost Notebook of S.R. Ramanujan was brought to light in 1976 as part of the Watson bequest, by G.E. Andrews with whose introduction this collection of unpublished manuscripts opens.

Author: Srinivasa Ramanujan Aiyangar

Publisher:

ISBN: 8185198063

Category: Fonctions thêta

Page: 419

View: 636

The so-called Lost Notebook of S.R. Ramanujan was brought to light in 1976 as part of the Watson bequest, by G.E. Andrews with whose introduction this collection of unpublished manuscripts opens. A major portion of the Lost Notebook - really just 90 unpaginated sheets of work on q-series and other topics - is reproduced here in facsimile. Letters from Ramanujan to Hardy as well as various other sheets of seemingly related notes are then included, on topics including coefficients in the 1/q3 and 1/q2 problems and the mock theta functions. The next 180 pages consist of unpublished manuscripts of Ramanujan, including 28 pages from the 'Loose Papers` held in the Trinity College Library. Finally a number of interesting letters that were exchanged between Ramanujan, Littlewood, Hardy and Watson, with a bearing on Ramanujan's work are collected together here with other extracts and fragments.

Srinivasa Ramanujan's achievements in research made him perhaps the greatest Indian mathematician of modern times.

Author: Srinivasa Ramanujan

Publisher:

ISBN: 8185198357

Category: Mathematics

Page: 419

View: 257

Srinivasa Ramanujan's achievements in research made him perhaps the greatest Indian mathematician of modern times. By the time he was 12 years old he had worked through a copy of Loney's Trigonometry. At the age of 16 he verified over 6,000 formulae in Carr's Synopsis of Pure Mathematics. By the time he was married at 21 he had already obtained several results in the areas of elliptic integrals, hypergeometric series and divergent series. In 1918 he was elected a Fellow of the Royal Society and a Fellow of Trinity College, but died shortly thereafter.

The work is likely to be of interest to those in number theory as well. The only required background is some knowledge of continued fractions and a course in complex analysis.

Author: George E. Andrews

Publisher: American Mathematical Soc.

ISBN: 9780821825389

Category: Mathematics

Page: 71

View: 439

Among his thirty-three published papers, Ramanujan had only one continued fraction, the Rogers-Ramanujan continued fraction. However, his notebooks contain over 100 results on continued fractions. At the end of his second notebook are 100 pages of unorganized material, and the third notebook comprises thirty-three pages of disorganized results. In these 133 pages of material are approximately sixty theorems on continued fractions, most of them new results. In this monograph, the authors discuss and prove each of these theorems. Aimed at those interested in Ramanujan and his work, this monograph will be of special interest to those who work in continued fractions, $q$-series, special functions, theta-functions, and combinatorics. The work is likely to be of interest to those in number theory as well. The only required background is some knowledge of continued fractions and a course in complex analysis.

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online.

Author: Source: Wikipedia

Publisher: Books LLC, Wiki Series

ISBN: 1233123467

Category:

Page: 26

View: 151

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 24. Chapters: Mock modular form, Ramanujan's sum, 1729, Tau-function, Taxicab number, Ramanujan's lost notebook, Ramanujan summation, A Disappearing Number, Ramanujan prime, Ramanujan graph, Ramanujan's congruences, Rogers-Ramanujan continued fraction, Rogers-Ramanujan identities, Ramanujan-Petersson conjecture, Ramanujan-Nagell equation, Six exponentials theorem, 4130 Ramanujan, Ramanujan theta function, The Man Who Knew Infinity, Ramanujan-Soldner constant, The Indian Clerk, Landau-Ramanujan constant, SASTRA Ramanujan Prize, ICTP Ramanujan Prize. Excerpt: Sr niv sa Aiyang r R m nujan FRS, better known as Srinivasa Iyengar Ramanujan ) (Tamil: ) (22 December 1887 - 26 April 1920) was an Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made substantial contributions to mathematical analysis, number theory, infinite series and continued fractions. Ramanujan's talent was said by the English mathematician G.H. Hardy to be in the same league as legendary mathematicians such as Gauss, Euler, Cauchy, Newton and Archimedes and is widely regarded as one of the towering geniuses in mathematics. Born in Erode, Tamil Nadu, India, to a poor Brahmin family, Ramanujan first encountered formal mathematics at age 10. He demonstrated a natural ability, and was given books on advanced trigonometry written by S. L. Loney. He mastered them by age 12, and even discovered theorems of his own, including independently re-discovering Euler's Identity. He demonstrated unusual mathematical skills at school, winning accolades and awards. By 17, Ramanujan conducted his own mathematical research on Bernoulli numbers and the Euler-Mascheroni constant. He received a scholarship to study at Government College in Kumbakonam, but lost it when he failed his non-mathematical coursework. He joined another college to pursue ...

In this paper, for 35 of the 40 identities, the authors offer proofs that are in the spirit of Ramanujan. Some of the proofs presented here are due to Watson, Rogers, and Bressoud, but most are new.

Author: Bruce C. Berndt

Publisher: American Mathematical Soc.

ISBN: 9780821839737

Category: Mathematics

Page: 96

View: 589

Sir Arthur Conan Doyle's famous fictional detective Sherlock Holmes and his sidekick Dr. Watson go camping and pitch their tent under the stars. During the night, Holmes wakes his companion and says, ""Watson, look up at the stars and tell me what you deduce."" Watson says, ""I see millions of stars, and it is quite likely that a few of them are planets just like Earth. Therefore there may also be life on these planets."" Holmes replies, ""Watson, you idiot. Somebody stole our tent."" When seeking proofs of Ramanujan's identities for the Rogers-Ramanujan functions, Watson, i.e., G. N. Watson, was not an ""idiot."" He, L. J. Rogers, and D. M. Bressoud found proofs for several of the identities. A. J. F. Biagioli devised proofs for most (but not all) of the remaining identities. Although some of the proofs of Watson, Rogers, and Bressoud are likely in the spirit of those found by Ramanujan, those of Biagioli are not. In particular, Biagioli used the theory of modular forms. Haunted by the fact that little progress has been made into Ramanujan's insights on these identities in the past 85 years, the present authors sought ""more natural"" proofs. Thus, instead of a missing tent, we have had missing proofs, i.e., Ramanujan's missing proofs of his forty identities for the Rogers-Ramanujan functions. In this paper, for 35 of the 40 identities, the authors offer proofs that are in the spirit of Ramanujan. Some of the proofs presented here are due to Watson, Rogers, and Bressoud, but most are new. Moreover, for several identities, the authors present two or three proofs. For the five identities that they are unable to prove, they provide non-rigorous verifications based on an asymptotic analysis of the associated Rogers-Ramanujan functions. This method, which is related to the 5-dissection of the generating function for cranks found in Ramanujan's lost notebook, is what Ramanujan might have used to discover several of the more difficult identities. Some of the new methods in this paper can be employed to establish new identities for the Rogers-Ramanujan functions.

All of the coefficients in Ramanujan's asymptotic expansion appear to be integers
, and this was recently proved by W. Galway ( 1 ) using a formula from Ramanujan's lost notebook ( 11 ) . Entry 17 , which can probably be generalized ,
gives ...

Author: Bruce C. Berndt

Publisher: Springer Science & Business Media

ISBN: 0387949410

Category: Mathematics

Page: 624

View: 482

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.

called ) ' Lost ' Notebook by Andrews ( 2 ) resulted in a resurgence of interest in Ramanujan's work . Now , at the beginning of this century , after completing the
editing of the 3254 Entries of Ramanujan in his Notebooks , Berndt and Andrews
...

Among his thirty-three published papers, Ramanujan had only one continued fraction, the Rogers-Ramanujan continued fraction. However, his notebooks contain over 100 results on continued fractions. At the end of his second notebook are 100 pages of unorganized material, and the third notebook comprises thirty-three pages of disorganized results. In these 133 pages of material are approximately sixty theorems on continued fractions, most of them new results. In this monograph, the authors discuss and prove each of these theorems. Aimed at those interested in Ramanujan and his work, this monograph will be of special interest to those who work in continued fractions, $q$-series, special functions, theta-functions, and combinatorics. The work is likely to be of interest to those in number theory as well. The only required background is some knowledge of continued fractions and a course in complex analysis.

Modular Equations in Ramanujan ' s Lost Notebook Bruce C . Berndt 0
Introduction Ramanujan recorded several hundred modular equations in his
three notebooks [ 7 ] ; no other mathematician has ever discovered nearly so
many . Complete ...

In 1976 he discovered Ramanujan's Lost Notebook. Besides giving the readers access to George Andrews' most important papers, the volume also provides his background commentary and comprehensive assessment.

Author: George E. Andrews

Publisher: Icp Selected Papers

ISBN: 1848166664

Category: Mathematics

Page: 1012

View: 739

George E Andrews is the Evan Pugh Professor of Mathematics at Pennsylvania. He is also the President of American Mathematical Society (AMS) during 2009 2011. George Andrews has been a world pioneer in partitions and q-series. His contributions include more than 250 scientific papers and several books on number theory and the theory of partitions. In 1976 he discovered Ramanujan's Lost Notebook. Besides giving the readers access to George Andrews' most important papers, the volume also provides his background commentary and comprehensive assessment.

[ 4 ] G. E. Andrews , “ Physics , Ramanujan , and computer algebra ” , as a Tool
for Research in Mathematics and Physics ... [ 5 ] G. E. Andrews and F. G. Garvan , Ramanujan's “ lost ” notebook VI : The mock theta conjectures , Adv . in Math .

In his very first letter to Hardy written from India , Ramanujan had included many
corollaries of this result by which Hardy confesses to have been “ defeated
completely ” . Ramanujan's Lost Notebook “ consisting of about 140 unpaginated
...

( BS ) B. C. Berndt and J. Sohn , Asymptotic formulas for two continued fractions
in Ramanujan's Lost Notebook , J. London Math . Soc . 65 ( 2002 ) , 271–284 . (
BY ] B. C. Berndt and A. J. Yee , On the generalized Rogers - Ramanujan ...

3. G.E. Andrews. On a trasnformation of bilateral series with applications, Proc.
Amer. Math. Soc. 25 (1970) 554-558. 4. G.E. Andrews. An introduction to Ramanujan's "lost" notebook, I, Partial 6-functions, Advances Math. 41 (1981)
137-172. 5.

On the occasion of the 100th anniversary of Ramanujan's birth, the Indian Prime
Minister Rajiv Gandhi announced the publication of Ramanujan's "lost notebook"
and presented the first copy to Ramanujan's widow. Andrew Wiles Andrew ...

Author: Harold R. Parks

Publisher:

ISBN: 0130116904

Category: Mathematics

Page: 814

View: 243

This contemporary approach to liberal arts math breaks away from traditional instruction and moves towards a more “modern” course that stresses rich ideas, little review, and more visualization. This readerfriendly book offers an accessible writing style and mathematical integrity. Its unique three-part organization (Life, Society, the World) presents readers with sound, relevant mathematics, leaving them with the (correct) impression that math is useful and affects their lives in many positive ways. Mathematical Structures and Methods. Descriptive Statistics. Collecting and Interpreting Data. Inferential Statistics. Probability. Consumer Mathematics. Management Mathematics. Critical Thinking, Logical Reasoning, and Problem Solving. Geometry. Growth and Scaling. For anyone who needs to learn or review basic math concepts and practical applications.

Author: American Mathematical SocietyPublish On: 2004

Emphasis will be given to q - continued fractions in Ramanujan's lost notebook .
Particular attention will focus upon " Ramanujan's continued fraction " and a
certain continued fraction of Ramanujan to base q3 . ( Received September 29 ,
2003 ) ...