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1991 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Ramanujan’s Notebooks

Part III

verfasst von: Bruce C. Berndt

Verlag: Springer New York

Enthalten in: Professional Book Archive

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter
Introduction
Abstract
The content of this volume is more unified than those of the first two volumes of our attempts to provide proofs of the many beautiful theorems bequeathed to us by Ramanujan in his notebooks. Theta-functions provide the binding glue that blends Chapters 16-21 together. Although we provide proofs here for all of Ramanujan’s formulas, in many cases, we have been unable to find the roads that led Ramanujan to his discoveries. It is hoped that others will attempt to discover the pathways that Ramanujan took on his journey through his luxuriant labyrinthine forest of enchanting and alluring formulas.
Bruce C. Berndt
Chapter 16. q-Series and Theta-Functions
Abstract
In Chapter 16, Ramanujan develops two closely related topics, q-series and theta-functions. The first 17 sections are devoted primarily to q-series, while the latter 22 sections constitute a very thorough development of the theory of theta-functions.
Bruce C. Berndt
Chapter 17. Fundamental Properties of Elliptic Functions
Abstract
Chapter 17 is almost entirely devoted to the theory of elliptic functions. The groundwork was prepared in the sections on theta-functions in Chapter 16. In the present chapter, Ramanujan introduces Jacobian elliptic functions and elliptic integrals. It is interesting that Ramanujan does not use the classical notation and terminology from the theory of elliptic functions and integrals. In Section 6, we identify the functions and parameters employed by Ramanujan with the more familiar notations in the theory of elliptic functions.
Bruce C. Berndt
Chapter 18. The Jacobian Elliptic Functions
Abstract
In Chapter 18, Ramanujan continues his development of the theory of elliptic functions begun in Chapter 16 with the theory of theta-functions and continued in Chapter 17 with an introduction to elliptic integrals and the compilation of a large catalog of series that can be evaluated in terms of elliptic function parameters. This chapter contains further series identities depending on the theory of elliptic functions. Such results are considerably fewer in number here than in Chapter 17 and generally are more difficult to prove. In particular, see Sections 4-7.
Bruce C. Berndt
Chapter 19. Modular Equations of Degrees 3, 5, and 7 and Associated Theta-Function Identities
Abstract
In several ways, this is a remarkable chapter. Not only are the results enormously interesting and often difficult to prove, but many questions arise in regard to Ramanujan’s methods of proof. Undoubtedly, many of the proofs given here are quite unlike those found by Ramanujan. He evidently possessed methods that we have been unable to discern. No hints whatsoever of his methods are provided by Ramanujan.
Bruce C. Berndt
Chapter 20. Modular Equations of Higher and Composite Degrees
Abstract
In this chapter, we continue to examine Ramanujan’s discoveries about modular equations. In the previous chapter, modular equations of degrees 3, 5, and 7 were derived. Modular equations of degrees 11, 13, 17, 19, 23, 31, 47, and 71 are established in this chapter. Also, modular equations of composite degree, or “mixed” modular equations, are studied. Most of the equations of the latter type involve four distinct moduli, and so we begin by defining such a modular equation.
Bruce C. Berndt
Chapter 21. Eisenstein Series
Abstract
Chapter 21 concludes the organized portion of Ramanujan’s second notebook; after Chapter 21, there are 100 pages of unorganized material. Chapter 21 constitutes only four pages and thus is the shortest chapter in the second notebook. Almost all of the previous chapters are twelve pages in length.
Bruce C. Berndt
Backmatter
Metadaten
Titel
Ramanujan’s Notebooks
verfasst von
Bruce C. Berndt
Copyright-Jahr
1991
Verlag
Springer New York
Electronic ISBN
978-1-4612-0965-2
Print ISBN
978-1-4612-6963-2
DOI
https://doi.org/10.1007/978-1-4612-0965-2