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Ramanujan's Lost Notebook de George E Andrews

Ramanujan's Lost Notebook

Autor George E Andrews, Bruce C Berndt
en Limba Engleză Hardback – 23 dec 2008

Aplicabilitatea practică a acestui volum rezidă în rigoarea cu care George E. Andrews și Bruce C. Berndt transformă intuițiile geniale, dar adesea laconice, ale lui Srinivasa Ramanujan într-un corpus matematic demonstrabil și utilizabil în cercetarea contemporană. Ne-a atras atenția modul în care autorii reușesc să organizeze materiale disparate — de la fragmente scrise în sanatorii între 1917 și 1919, până la scrisori adresate lui G. H. Hardy — într-o structură coerentă de 16 capitole. Această ediție din 2009 nu este doar o reproducere, ci o examinare critică a 314 intrări matematice, oferind dovezi acolo unde Ramanujan a lăsat doar afirmații.

Remarcăm progresia tematică a cuprinsului, care debutează cu transformările Heine și Sears–Thomae, avansând spre o analiză exhaustivă a fracțiilor continue Rogers-Ramanujan și a proprietăților lor modulare. Cartea reprezintă o alternativă la Ramanujan's Theta Functions de Shaun Cooper pentru cursurile de geometrie algebrică sau teoria numerelor, cu avantajul că Ramanujan's Lost Notebook oferă o perspectivă istorică și analitică asupra surselor primare, nu doar o dezvoltare sistematică a teoriei. Dacă volumul lui Cooper este un instrument pedagogic pentru studenți avansați, lucrarea de față este un document de referință pentru cercetători, documentând procesul de „recuperare” a unor identități matematice care au influențat profund combinatorica și funcțiile speciale. Stilul este unul tehnic, extrem de amănunțit, unde fiecare capitol beneficiază de o introducere contextuală necesară înțelegerii fragmentelor originale.

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Specificații

ISBN-13: 9780387777658
ISBN-10: 0387777652
Pagini: 420
Ilustrații: XII, 420 p. 8 illus.
Dimensiuni: 161 x 241 x 27 mm
Greutate: 0.86 kg
Ediția:2009 edition
Editura: Springer
Locul publicării:New York, NY, United States

Public țintă

Research

De ce să citești această carte

Recomandăm această lucrare cercetătorilor în matematică pură și istoria matematicii care doresc să exploreze moștenirea lui Ramanujan dincolo de rezultatele publicate antum. Cititorul câștigă acces la demonstrații moderne pentru celebrele identități Rogers-Ramanujan și o înțelegere profundă a fracțiilor continue. Este un instrument indispensabil pentru bibliotecile universitare, oferind puntea necesară între intuiția istorică și rigoarea matematică actuală.

Descriere scurtă

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." The "lost notebook" contains considerable material on mock theta functions and so undoubtedly emanates from the last year of Ramanujan's life. It should be emphasized that the material on mock theta functions is perhaps Ramanujan's deepest work.

Descriere

This is the second of approximately four volumes that the authors plan to write in their examination of all the claims made by S. Ramanujan in The Lost Notebook and Other Unpublished Papers. This volume, published by Narosa in 1988, contains the “Lost Notebook,” which was discovered by the ?rst author in the spring of 1976 at the library of Trinity College, Cambridge. Also included in this publication are other partial manuscripts, fragments, and letters that Ramanujan wrote to G. H. Hardy from nursing homes during 1917–1919. The authors have attempted to organize this disparate material in chapters. This second volume contains 16 chapters comprising 314 entries, including some duplications and examples, with chapter totals ranging from a high of ?fty-four entries in Chapter 1 to a low of two entries in Chapter 12. Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 The Heine Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1. 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1. 2 Heine’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1. 3 Ramanujan’s Proof of the q-Gauss Summation Theorem . . . . . 10 1. 4 Corollaries of (1. 2. 1) and (1. 2. 5) . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1. 5 Corollaries of (1. 2. 6) and (1. 2. 7) . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1. 6 Corollaries of (1. 2. 8), (1. 2. 9), and (1. 2. 10) . . . . . . . . . . . . . . . . . . 24 1. 7 Corollaries of Section 1. 2 and Auxiliary Results . . . . . . . . . . . . . 27 2 The Sears–Thomae Transformation . . . . . . . . . . . . . . . . . . . . . . . . 45 2. 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2. 2 Direct Corollaries of (2. 1. 1) and (2. 1. 3) . . . . . . . . . . . . . . . . . . . . 45 2. 3 Extended Corollaries of (2. 1. 1) and (2. 1. 3) . . . . . . . . . . . . . . . . .

Cuprins

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.

Recenzii

From the reviews: “This volume contains 16 chapters comprising 314 entries. The material is arranged thematically with the main topics being some of Ramanujan’s favorites q series theta functions … . the authors treatment is extremely thorough. Each chapter contains an introduction with appropriate background. References to all other known proofs of the entries are provided. … Fans of Ramanujan’s mathematics are sure to be delighted by this book. … Many entries are just begging for further study and will undoubtedly be inspiring research for decades to come.” (Jeremy Lovejoy, Mathematical Reviews, Issue 2010 f)

Textul de pe ultima copertă

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

Caracteristici

Most of this material has never before been published in book form
Authors have organized, and provided commentary on, Ramanujan's results
Discusses q-series, Eisenstein series, and theta functions
Includes letters to G.H. Hardy