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Continued Fractions with Applications

Author : L. Lorentzen

Publisher : North Holland

Page : 634 pages

File Size : 40,31 MB

Release : 1992-11-08

Category : Computers

ISBN :

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Book Description

This book is aimed at two kinds of readers: firstly, people working in or near mathematics, who are curious about continued fractions; and secondly, senior or graduate students who would like an extensive introduction to the analytic theory of continued fractions. The book contains several recent results and new angles of approach and thus should be of interest to researchers throughout the field. The first five chapters contain an introduction to the basic theory, while the last seven chapters present a variety of applications. Finally, an appendix presents a large number of special continued fraction expansions. This very readable book also contains many valuable examples and problems.



CONTINUED FRACTIONS

Author : Haakon Waadeland

Publisher : Springer Science & Business Media

Page : 321 pages

File Size : 21,65 MB

Release : 2008-04-01

Category : Mathematics

ISBN : 9491216376

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Book Description

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.



Continued Fractions

Author : Aleksandr I?Akovlevich Khinchin

Publisher : Courier Corporation

Page : 116 pages

File Size : 26,26 MB

Release : 1997-05-14

Category : Mathematics

ISBN : 9780486696300

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Book Description

Elementary-level text by noted Soviet mathematician offers superb introduction to positive-integral elements of theory of continued fractions. Clear, straightforward presentation of the properties of the apparatus, the representation of numbers by continued fractions, and the measure theory of continued fractions. 1964 edition. Prefaces.



Geometry of Continued Fractions

Author : Oleg Karpenkov

Publisher : Springer Science & Business Media

Page : 409 pages

File Size : 42,85 MB

Release : 2013-08-15

Category : Mathematics

ISBN : 3642393683

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Book Description

Traditionally a subject of number theory, continued fractions appear in dynamical systems, algebraic geometry, topology, and even celestial mechanics. The rise of computational geometry has resulted in renewed interest in multidimensional generalizations of continued fractions. Numerous classical theorems have been extended to the multidimensional case, casting light on phenomena in diverse areas of mathematics. This book introduces a new geometric vision of continued fractions. It covers several applications to questions related to such areas as Diophantine approximation, algebraic number theory, and toric geometry. The reader will find an overview of current progress in the geometric theory of multidimensional continued fractions accompanied by currently open problems. Whenever possible, we illustrate geometric constructions with figures and examples. Each chapter has exercises useful for undergraduate or graduate courses.



Metrical Theory of Continued Fractions

Author : M. Iosifescu

Publisher : Springer Science & Business Media

Page : 408 pages

File Size : 13,15 MB

Release : 2002-09-30

Category : Mathematics

ISBN : 9781402008924

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Book Description

The book is essentially based on recent work of the authors. In order to unify and generalize the results obtained so far, new concepts have been introduced, e.g., an infinite order chain representation of the continued fraction expansion of irrationals, the conditional measures associated with, and the extended random variables corresponding to that representation. Also, such procedures as singularization and insertion allow to obtain most of the continued fraction expansions related to the regular continued fraction expansion. The authors present and prove with full details for the first time in book form, the most recent developments in solving the celebrated 1812 Gauss' problem which originated the metrical theory of continued fractions. At the same time, they study exhaustively the Perron-Frobenius operator, which is of basic importance in this theory, on various Banach spaces including that of functions of bounded variation on the unit interval. The book is of interest to research workers and advanced Ph.D. students in probability theory, stochastic processes and number theory.



Continued Fractions and Orthogonal Functions

Author : S. Clement Cooper

Publisher : CRC Press

Page : 402 pages

File Size : 31,95 MB

Release : 1993-11-17

Category : Mathematics

ISBN : 9780824790714

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Book Description

This reference - the proceedings of a research conference held in Loen, Norway - contains information on the analytic theory of continued fractions and their application to moment problems and orthogonal sequences of functions. Uniting the research efforts of many international experts, this volume: treats strong moment problems, orthogonal polynomials and Laurent polynomials; analyses sequences of linear fractional transformations; presents convergence results, including truncation error bounds; considers discrete distributions and limit functions arising from indeterminate moment problems; discusses Szego polynomials and their applications to frequency analysis; describes the quadrature formula arising from q-starlike functions; and covers continued fractional representations for functions related to the gamma function.;This resource is intended for mathematical and numerical analysts; applied mathematicians; physicists; chemists; engineers; and upper-level undergraduate and agraduate students in these disciplines.



Continued Fractions and Signal Processing

Author : Tomas Sauer

Publisher : Springer Nature

Page : 275 pages

File Size : 16,60 MB

Release : 2021-09-06

Category : Mathematics

ISBN : 3030843602

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Book Description

Besides their well-known value in number theory, continued fractions are also a useful tool in modern numerical applications and computer science. The goal of the book is to revisit the almost forgotten classical theory and to contextualize it for contemporary numerical applications and signal processing, thus enabling students and scientist to apply classical mathematics on recent problems. The books tries to be mostly self-contained and to make the material accessible for all interested readers. This provides a new view from an applied perspective, combining the classical recursive techniques of continued fractions with orthogonal problems, moment problems, Prony’s problem of sparse recovery and the design of stable rational filters, which are all connected by continued fractions.



Neverending Fractions

Author : Jonathan Borwein

Publisher : Cambridge University Press

Page : 223 pages

File Size : 23,80 MB

Release : 2014-07-03

Category : Mathematics

ISBN : 0521186498

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Book Description

This introductory text covers a variety of applications to interest every reader, from researchers to amateur mathematicians.



Continued Fractions

Author : Doug Hensley

Publisher : World Scientific

Page : 261 pages

File Size : 10,52 MB

Release : 2006-03-01

Category : Mathematics

ISBN : 9814479438

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Book Description

The Euclidean algorithm is one of the oldest in mathematics, while the study of continued fractions as tools of approximation goes back at least to Euler and Legendre. While our understanding of continued fractions and related methods for simultaneous diophantine approximation has burgeoned over the course of the past decade and more, many of the results have not been brought together in book form. Continued fractions have been studied from the perspective of number theory, complex analysis, ergodic theory, dynamic processes, analysis of algorithms, and even theoretical physics, which has further complicated the situation.This book places special emphasis on continued fraction Cantor sets and the Hausdorff dimension, algorithms and analysis of algorithms, and multi-dimensional algorithms for simultaneous diophantine approximation. Extensive, attractive computer-generated graphics are presented, and the underlying algorithms are discussed and made available.



Handbook of Continued Fractions for Special Functions

Author : Annie A.M. Cuyt

Publisher : Springer Science & Business Media

Page : 430 pages

File Size : 38,19 MB

Release : 2008-04-12

Category : Mathematics

ISBN : 1402069499

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Book Description

Special functions are pervasive in all fields of science and industry. The most well-known application areas are in physics, engineering, chemistry, computer science and statistics. Because of their importance, several books and websites (see for instance http: functions.wolfram.com) and a large collection of papers have been devoted to these functions. Of the standard work on the subject, the Handbook of mathematical functions with formulas, graphs and mathematical tables edited by Milton Abramowitz and Irene Stegun, the American National Institute of Standards claims to have sold over 700 000 copies! But so far no project has been devoted to the systematic study of continued fraction representations for these functions. This handbook is the result of such an endeavour. We emphasise that only 10% of the continued fractions contained in this book, can also be found in the Abramowitz and Stegun project or at the Wolfram website!