Ramanujan's Notebooks Volume 4

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. Booknews "Ramanujan studies" have recently--and happily--become something of an industry; several mathematicians are presently devoting their professional energy to the explication of inspired insight and unearthly beauty long buried in Ramanujan's notebooks (themselves recently published, in photo-facsimile, for the first time). The present volume represents the continuation of a projected four-part series, by a mathematician associated with the University of Illinois. In six chapters it treats notebook entries pertaining to hypergeometric series, continued fractions, integrals and asymptotic expansions, infinite series, asymptotic expansions and modular forms. Chapters begin with an introductory essay, followed by notebook entries, proofs, corollaries and remarks, an overall objective being to detect and expose the pattern of Ramanujan's argument (at this point still frequently elusive). (NW) Annotation c. Book News, Inc., Portland, OR (booknews.com)

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 Tasks 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.

booknews

ramanujan Studies Have Recently--and Happily--become Something Of An Industry; Several Mathematicians Are Presently Devoting Their Professional Energy To The Explication Of Inspired Insight And Unearthly Beauty Long Buried In Ramanujan's Notebooks (themselves Recently Published, In Photo-facsimile, For The First Time). The Present Volume Represents The Continuation Of A Projected Four-part Series, By A Mathematician Associated With The University Of Illinois. In Six Chapters It Treats Notebook Entries Pertaining To Hypergeometric Series, Continued Fractions, Integrals And Asymptotic Expansions, Infinite Series, Asymptotic Expansions And Modular Forms. Chapters Begin With An Introductory Essay, Followed By Notebook Entries, Proofs, Corollaries And Remarks, An Overall Objective Being To Detect And Expose The Pattern Of Ramanujan's Argument (at This Point Still Frequently Elusive). (nw) Annotation C. Book News, Inc., Portland, Or (booknews.com)

Pt. 1. Magic Squares. Sums Related To The Harmonic Series Or The Inverse Tangent Function. Combinatorial Analysis And Series Inversions. Iterates Of The Exponential Function And An Ingenious Formal Technique. Eluerian Polynomials And Numbers, Bernoulli Numbers, And The Riemann Zeta-function. Ramanujan's Theory Of Divergent Series. Sums Of Powers, Bernoulli Numbers, And The Gamma Function. Analogues Of The Gamma Function. Infinite Series Identities, Transformations, And Evaluations. Ramanujan's Quarterly Reports -- Pt. 2. Hypergeometric Series, I. Hypergeometric Series, Ii. Continued Fractions. Integrals And Asymptotic Expansions. Infinite Series. Asymptotic Expansions And Modular Forms -- Pt. 3. Q-series And Theta-functions. Fundamental Properties Of Elliptic Functions. The Jacobian Elliptic Functions. Modular Equations Of Degrees 3, 5, And 7 And Associated Theta-function Identities. Modular Equations Of Higher And Composite Degrees. Eisenstein Series [edited By] Bruce C. Berndt. Includes Index. Bibliography: V. 1, P. [337]-352. 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. 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. During the years 1903-1914, Ramanujan recorded many of his mathematical discoveries in notebooks without providing proofs. 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. In this chapter we will examine Ramanujan's findings that require only elementary high-school algebra for an understanding.