The Continued Fractions Found In The Unorganized Portions Of Ramanujan S Notebooks

Author : George E. Andrews
Publisher : American Mathematical Soc.
Page : 82 pages
File Size : 45,7 Mb
Release : 1992
Category : Continued fractions
ISBN : 9780821825389

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The Continued Fractions Found in the Unorganized Portions of Ramanujan's Notebooks by George E. Andrews Pdf

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 function, 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.

Author : Haakon Waadeland,Lisa Lorentzen
Publisher : Springer Science & Business Media
Page : 308 pages
File Size : 54,7 Mb
Release : 2008-04-01
Category : Mathematics
ISBN : 9789491216374

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CONTINUED FRACTIONS by Haakon Waadeland,Lisa Lorentzen Pdf

Continued Fractions consists of two volumes — Volume 1: Convergence Theory; and Volume 2: Representation of Functions (tentative title), which is expected in 2011. Volume 1 is dedicated to the convergence and computation of continued fractions, while Volume 2 will treat representations of meromorphic functions by continued fractions. Taken together, the two volumes will present the basic continued fractions theory without requiring too much previous knowledge; some basic knowledge of complex functions will suffice. Both new and advanced graduate students of continued fractions shall get a comprehensive understanding of how these infinite structures work in a number of applications, and why they work so well. A varied buffet of possible applications to whet the appetite is presented first, before the more basic but modernized theory is given. This new edition is the result of an increasing interest in computing special functions by means of continued fractions. The methods described in detail are, in many cases, very simple, yet reliable and efficient.

Author : Krishnaswami Alladi
Publisher : Springer
Page : 240 pages
File Size : 43,8 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540466819

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Number Theory, Madras 1987 by Krishnaswami Alladi Pdf

Author : Srinivasa Ramanujan Aiyangar
Publisher : American Mathematical Soc.
Page : 366 pages
File Size : 40,5 Mb
Release : 1995-09-07
Category : Mathematics
ISBN : 0821891251

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Ramanujan by Srinivasa Ramanujan Aiyangar Pdf

The letters that Ramanujan wrote to G. H. Hardy on January 16 and February 27, 1913, are two of the most famous letters in the history of mathematics. These and other letters introduced Ramanujan and his remarkable theorems to the world and stimulated much research, especially in the 1920s and 1930s. This book brings together many letters to, from, and about Ramanujan. The letters came from the National Archives in Delhi, the Archives in the State of Tamil Nadu, and a variety of other sources. Helping to orient the reader is the extensive commentary, both mathematical and cultural, by Berndt and Rankin; in particular, they discuss in detail the history, up to the present day, of each mathematical result in the letters. Containing many letters that have never been published before, this book will appeal to those interested in Ramanujan's mathematics as well as those wanting to learn more about the personal side of his life. Ramanujan: Letters and Commentary was selected for the CHOICE list of Outstanding Academic Books for 1996.

Author : Lisa Lorentzen,Haakon Waadeland
Publisher : atlantis press
Page : 321 pages
File Size : 51,6 Mb
Release : 2008
Category : Mathematics
ISBN : 9789078677079

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Continued Fractions by Lisa Lorentzen,Haakon Waadeland Pdf

Continued Fractions consists of two volumes -- Volume 1: Convergence Theory; and Volume 2: Representation of Functions (tentative title), which is expected in 2011. Volume 1 is dedicated to the convergence and computation of continued fractions, while Volume 2 will treat representations of meromorphic functions by continued fractions. Taken together, the two volumes will present the basic continued fractions theory without requiring too much previous knowledge; some basic knowledge of complex functions will suffice. Both new and advanced graduate students of continued fractions shall get a comprehensive understanding of how these infinite structures work in a number of applications, and why they work so well. A varied buffet of possible applications to whet the appetite is presented first, before the more basic but modernized theory is given.This new edition is the result of an increasing interest in computing special functions by means of continued fractions. The methods described in detail are, in many cases, very simple, yet reliable and efficient.

Author : A.K. Agarwal,Bruce C. Berndt,Christian F. Krattenthaler,Gary L. Mullen,K. Ramachandra,Michel Waldschmidt
Publisher : Birkhäuser
Page : 314 pages
File Size : 54,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034882231

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Number Theory and Discrete Mathematics by A.K. Agarwal,Bruce C. Berndt,Christian F. Krattenthaler,Gary L. Mullen,K. Ramachandra,Michel Waldschmidt Pdf

To mark the World Mathematical Year 2000 an International Conference on Number Theory and Discrete Mathematics in honour of the legendary Indian Mathematician Srinivasa Ramanuj~ was held at the centre for Advanced study in Mathematics, Panjab University, Chandigarh, India during October 2-6, 2000. This volume contains the proceedings of that conference. In all there were 82 participants including 14 overseas participants from Austria, France, Hungary, Italy, Japan, Korea, Singapore and the USA. The conference was inaugurated by Prof. K. N. Pathak, Hon. Vice-Chancellor, Panjab University, Chandigarh on October 2, 2000. Prof. Bruce C. Berndt of the University of Illinois, Urbana Chaimpaign, USA delivered the key note address entitled "The Life, Notebooks and Mathematical Contributions of Srinivasa Ramanujan". He described Ramanujan--as one of this century's most influential Mathematicians. Quoting Mark K. ac, Prof. George E. Andrews of the Pennsylvania State University, USA, in his message for the conference, described Ramanujan as a "magical genius". During the 5-day deliberations invited speakers gave talks on various topics in number theory and discrete mathematics. We mention here a few of them just as a sampling: • M. Waldschmidt, in his article, provides a very nice introduction to the topic of multiple poly logarithms and their special values. • C.

Author : Bruce C. Berndt
Publisher : American Mathematical Soc.
Page : 388 pages
File Size : 42,6 Mb
Release : 2001
Category : Biography & Autobiography
ISBN : 0821826247

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Ramanujan: Essays and Surveys by Bruce C. Berndt Pdf

This book contains essays on Ramanujan and his work that were written especially for this volume. It also includes important survey articles in areas influenced by Ramanujan's mathematics. Most of the articles in the book are nontechnical, but even those that are more technical contain substantial sections that will engage the general reader. The book opens with the only four existing photographs of Ramanujan, presenting historical accounts of them and information about other people in the photos. This section includes an account of a cryptic family history written by his younger brother, S. Lakshmi Narasimhan. Following are articles on Ramanujan's illness by R. A. Rankin, the British physician D. A. B. Young, and Nobel laureate S. Chandrasekhar. They present a study of his symptoms, a convincing diagnosis of the cause of his death, and a thorough exposition of Ramanujan's life as a patient in English sanitariums and nursing homes. Following this are biographies of S. Janaki (Mrs. Ramanujan) and S. Narayana Iyer, Chief Accountant of the Madras Port Trust Office, who first communicated Ramanujan's work to the Journal of the Indian Mathematical Society. The last half of the book begins with a section on ``Ramanujan's Manuscripts and Notebooks''. Included is an important article by G. E. Andrews on Ramanujan's lost notebook. The final two sections feature both nontechnical articles, such as Jonathan and Peter Borwein's ``Ramanujan and pi'', and more technical articles by Freeman Dyson, Atle Selberg, Richard Askey, and G. N. Watson. This volume complements the book Ramanujan: Letters and Commentary, Volume 9, in the AMS series, History of Mathematics. For more on Ramanujan, see these AMS publications Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, Volume 136.H, and Collected Papers of Srinivasa Ramanujan, Volume 159.H, in the AMS Chelsea Publishing series.

Author : Bruce C. Berndt
Publisher : Springer Science & Business Media
Page : 630 pages
File Size : 48,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461216247

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Ramanujan’s Notebooks by Bruce C. Berndt Pdf

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.

Author : Jorge Ize,Ivar Massabo,Alfonso Vignoli
Publisher : American Mathematical Soc.
Page : 179 pages
File Size : 54,9 Mb
Release : 1992
Category : Mathematics
ISBN : 9780821825426

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Degree Theory for Equivariant Maps, the General $S^1$-Action by Jorge Ize,Ivar Massabo,Alfonso Vignoli Pdf

This work is devoted to a detailed study of the equivariant degree and its applications for the case of an $S^1$-action. This degree is an element of the equivariant homotopy group of spheres, which are computed in a step-by-step extension process. Applications include the index of an isolated orbit, branching and Hopf bifurcation, and period doubling and symmetry breaking for systems of autonomous differential equations. The authors have paid special attention to making the text as self-contained as possible, so that the only background required is some familiarity with the basic ideas of homotopy theory and of Floquet theory in differential equations. Illustrating in a natural way the interplay between topology and analysis, this book will be of interest to researchers and graduate students.

Author : Robert Gordon
Publisher : American Mathematical Soc.
Page : 129 pages
File Size : 45,6 Mb
Release : 1993
Category : Mathematics
ISBN : 9780821825433

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G-categories by Robert Gordon Pdf

A $G$-category is a category on which a group $G$ acts. This work studies the $2$-category $G$-Cat of $G$-categories, $G$-functors (functors which commute with the action of $G$ ) and $G$-natural transformations (natural transformations which commute with the $G$-action). There is particular emphasis on the relationship between a $G$-category and its stable subcategory, the largest sub-$G$-category on which $G$ operates trivially. Also contained here are some very general applications of the theory to various additive $G$-categories and to $G$-topoi.

Author : Anonim
Publisher : Unknown
Page : 634 pages
File Size : 51,9 Mb
Release : 2006
Category : Mathematics
ISBN : UOM:39015068677874

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Advanced Studies in Contemporary Mathematics by Anonim Pdf

Author : Anthony A. Iarrobino
Publisher : American Mathematical Soc.
Page : 115 pages
File Size : 42,5 Mb
Release : 1994
Category : Mathematics
ISBN : 9780821825761

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Associated Graded Algebra of a Gorenstein Artin Algebra by Anthony A. Iarrobino Pdf

In 1904, Macaulay described the Hilbert function of the intersection of two plane curve branches: It is the sum of a sequence of functions of simple form. This monograph describes the structure of the tangent cone of the intersection underlying this symmetry. Iarrobino generalizes Macaulay's result beyond complete intersections in two variables to Gorenstein Artin algebras in an arbitrary number of variables. He shows that the tangent cone of a Gorenstein singularity contains a sequence of ideals whose successive quotients are reflexive modules. Applications are given to determining the multiplicity and orders of generators of Gorenstein ideals and to problems of deforming singular mapping germs. Also included are a survey of results concerning the Hilbert function of Gorenstein Artin algebras and an extensive bibliography.

Author : Svante Janson
Publisher : American Mathematical Soc.
Page : 78 pages
File Size : 55,5 Mb
Release : 1994
Category : Mathematics
ISBN : 9780821825952

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Orthogonal Decompositions and Functional Limit Theorems for Random Graph Statistics by Svante Janson Pdf

This book develops a method to obtain limit theorems for various functionals of random graphs. The method is based on a certain orthogonal decomposition. Janson's results include limit theorems for the two standard random graph models, $G_{n,p}$ and $G_{n,m}$, as well as functional limit theorems for the evolution of a random graph and results on the maximum of a function during the evolution. Janson obtains both normal and nonnormal limits, and the method provides an explanation for the appearance of nonnormal limits. Applications to subgraph counts and to vertex degrees are presented as examples.

Author : Henry C. Wente
Publisher : American Mathematical Soc.
Page : 77 pages
File Size : 44,6 Mb
Release : 1992
Category : Mathematics
ISBN : 9780821825365

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Constant Mean Curvature Immersions of Enneper Type by Henry C. Wente Pdf

This memoir is devoted to the case of constant mean curvature surfaces immersed in [bold]R3. We reduce this geometrical problem to finding certain integrable solutions to the Gauss equation. Many new and interesting examples are presented, including immersed cylinders in [bold]R3 with embedded Delaunay ends and [italic]n-lobes in the middle, and one-parameter families of immersed constant mean curvature tori in [bold]R3. We examine minimal surfaces in hyperbolic three-space, which is in some ways the most complicated case.

Author : George Allan Hagedorn
Publisher : American Mathematical Soc.
Page : 130 pages
File Size : 47,9 Mb
Release : 1994
Category : Mathematics
ISBN : 9780821826058

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Molecular Propagation through Electron Energy Level Crossings by George Allan Hagedorn Pdf

The principal results of this paper involve the extension of the time-dependent Born-Oppenheimer approximation to accommodate the propagation of nuclei through generic, minimal multiplicity electron energy level crossings. The Born-Oppenheimer approximation breaks down at electron energy level crossings, which are prevalent in molecular systems. We classify generic, minimal multiplicity level crossings and derives a normal form for the electron Hamiltonian near each type of crossing. We then extend the time-dependent Born-Oppenheimer approximation to accommodate the propagation of nuclei through each type of electron energy level crossing.