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Ramanujan’s Notebooks

Part V

verfasst von: Bruce C. Berndt

Verlag: Springer New York

Enthalten in: Professional Book Archive

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During the years 1903-1914, Ramanujan recorded most of his mathematical dis coveries without proofs in notebooks. Although many of his results had already been published by others, most had not. Almost a decade after Ramanujan's death in 1920, G. N. Watson and B. M. Wilson began to edit Ramanujan's notebooks, but, despite devoting over ten years to this project, they never completed their task. An unedited photostat edition of the notebooks was published by the Tata Institute of Fundamental Research in Bombay in 1957. This book is the fifth and final volume devoted to the editing of Ramanujan's notebooks. Parts I-III, published, respectively, in 1985, 1989, and 1991, contain accounts of Chapters 1-21 in the second notebook, a revised enlarged edition of the first. Part IV, published in 1994, contains results from the 100 unorganized pages in the second notebook and the 33 unorganized pages comprising the third notebook. Also examined in Part IV are the 16 organized chapters in the first notebook, which contain very little that is not found in the second notebook. In this fifth volume, we examine the remaining contents from the 133 unorganized pages in the second and third notebooks, and the claims in the 198 unorganized pages of the first notebook that cannot be found in the succeeding notebooks.

Inhaltsverzeichnis

Frontmatter

Introduction

Abstract

This book constitutes the fifth and final volume of our attempts to establish all the results claimed by the great Indian mathematician Srinivasa Ramanujan in hisNotebooksfirst published in a photostat edition by the Tata Institute of Fundamental Research in 1957 [9]. Although each of the five volumes contains many deep results, perhaps the average depth in this volume is greater than in the first four. As will be seen in the following paragraphs, several mathematicians made important contributions to the completion of this volume. However, I particularly extend my deepest gratitude to Heng Huat Chan and Liang—Cheng Zhang without whose contributions this volume would have been woefully deficient. This volume, however, should not be regarded as the closing chapter on Ramanujan’s notebooks. Instead, it is just the first milestone on our journey to understanding Ramanujan’s ideas. Many of the proofs given here and in other volumes certainly do not reflect Ramanujan’s motivation, insights, proofs, and wisdom. It is our fervent wish that these volumes will serve as springboards for further investigations by mathematicians intrigued by Ramanujan’s remarkable ideas. As in the other four volumes, for each correct claim, we either provide a proof or cite references in the literature where proofs can be found. We emphasize that Ramanujan made extremely few errors, and that most “mistakes” are either minor misprints, or, in fact, they are errors made by the author arising from misinterpretations of Ramanujan’s claims, which are occasionally fuzzy.

Bruce C. Berndt

32. Continued Fractions

Abstract

Chapter 12 in Ramanujan’s second notebook is devoted almost entirely to continued fractions. Further continued fractions can be found in other chapters, especially in Chapter 16. See Parts II [2] and III [3] for accounts of Chapters 12 and 16, respectively. The 100 pages of unorganized material at the end of the second notebook and the 33 unorganized pages in the third notebook contain about 60 further results on continued fractions. These and four evaluations of the RogersRamanujan continued fraction from the first notebook will be examined in this chapter.

Bruce C. Berndt

33. Ramanujan’s Theories of Elliptic Functions to Alternative Bases

Abstract

In his famous paper [3], [10, pp. 23–39], Ramanujan offers several beautiful series representations for 1/pi. He first states three formulas, one of which is

$$ \frac{4}{\pi } = \sum\limits_{{\mathbf{n = 0}}}^\infty {\frac{{\left( {6{\mathbf{n}} + 1} \right){{\left( {\frac{1}{2}} \right)}^3}_n}}{{{{\left( {{\mathbf{n}}!} \right)}^3}{4^n}}}} $$

where (a)o = 1 and, for each positive integer n

$$ {\left( {\mathbf{a}} \right)_n} = {\mathbf{a}}\left( {{\mathbf{a}} + 1} \right)\left( {{\mathbf{a}} + 2} \right)...\left( {{\mathbf{a}} + {\mathbf{n}} - 1} \right) $$

Bruce C. Berndt

34. Class Invariants and Singular Moduli

Abstract

So that we may define Ramanujan’sclass invariants, set

$$ (a;q)_\infty = \prod\limits_{n = 0}^\infty {(1 - aq^n )} ,\left| q \right| < 1, $$

and

$$ \chi (q) = ( - q;q^2 )_\infty $$

(1.1)

Bruce C. Berndt

35. Values of Theta—Functions

Abstract

For the convenience of the reader we briefly review here some definitions from earlier chapters.

Bruce C. Berndt

36. Modular Equations and Theta—Function Identities in Notebook 1

Abstract

Chapters 19-21 in Ramanujan’s second notebook are devoted almost exclusively to modular equations (Part III [3, pp. 220–488]). Ramanujan clearly loved modular equations, and, as the content of Chapters 34 and 35 makes manifest, he found many applications of these equations. Thus, it is surprising that the first notebook contains several dozen modular equations that he failed to record in his second notebook. Some of these are easy to prove with the help of modular equations in the second notebook, and so Ramanujan might have considered them less important and not worthy of repeating in his second notebook. However, many of them are apparently not so easy to prove. Some have degrees not examined in the second notebook. For example, on page 298 Ramanujan records a modular equation of degree 49. Not only does Ramanujan not consider modular equations of this degree in his second notebook, but apparently no one else had found a modular equation of degree 49 up until that time. Even at this writing, we know of no other modular equation of degree 49.

Bruce C. Berndt

37. Infinite Series

Abstract

In the last two chapters devoted to the second and third notebooks, we gather together most of Ramanujan’s results on series in the 133 pages of unorganized material found in these two notebooks. In this chapter, we primarily focus on exact formulas, while in Chapter 38 our attention is given to approximations and asymptotic formulas. In Part IV [4], we had disengaged Ramanujan’s results on special functions, partial fraction decompositions, and elementary and miscellaneous analysis from the material on infinite series, and devoted individual chapters to these three topics. Although those three chapters contain a couple of gems, Chapters 37 and 38 have many more jewels.

Bruce C. Berndt

38. Approximations and Asymptotic Expansions

Abstract

One of the primary areas to which Ramanujan made fundamental contributions, but for which he received no recognition until recent times, is asymptotic analysis. Asymptotic formulas, both general and specific, can be found in several places in his second notebook, but perhaps the largest concentration lies in Chapter 13. Several contributions pertain to hypergeometric functions, and an excellent survey of several of these results has been made by R. J. Evans [1]. The unorganized pages in the second and third notebooks also contain many beautiful theorems in asymptotic analysis. This chapter is devoted to proving these theorems and a few approximations as well.

Bruce C. Berndt

39. Miscellaneous Results in the First Notebook

Abstract

In this last chapter we collect together some miscellaneous results from the unorganized portions of the first notebook. Most are from analysis, with some pertaining to hypergeometric functions.

Bruce C. Berndt

Backmatter

Titel: Ramanujan’s Notebooks
verfasst von: Bruce C. Berndt
Verlag: Springer New York
Electronic ISBN: 978-1-4612-1624-7
Print ISBN: 978-1-4612-7221-2
DOI: https://doi.org/10.1007/978-1-4612-1624-7

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Inhaltsverzeichnis

Frontmatter

Introduction

32. Continued Fractions

33. Ramanujan’s Theories of Elliptic Functions to Alternative Bases

34. Class Invariants and Singular Moduli

35. Values of Theta—Functions

36. Modular Equations and Theta—Function Identities in Notebook 1

37. Infinite Series

38. Approximations and Asymptotic Expansions

39. Miscellaneous Results in the First Notebook

Backmatter