Handbook of Continued Fractions for Special Functions 2008th Edition by Annie A.M. Cuyt (PDF)

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!

User’s Reviews

Editorial Reviews: Review From the reviews:”The book under review is to collect continued fractions for special functions in a friendly volume. … There is no doubt that this book is very useful for the people who need to work with special functions. … The book is suitable for researchers in many fields, so that libraries serving scholars in basic sciences need to have it.” (Mehdi Hassani, The Mathematical Association of America, September, 2008) “The computation of a special function by its development into a continued fraction requires many steps, and that it needs a deep knowledge of theory and practice. The purpose of this handbook is to present the efforts in this direction made by a group of scientists from different universities who have collaborated in the project for many years. … I highly recommend this monograph, a masterpiece, to all those who are interested in continued fractions. It should be on the shelves of every library.” (Claude Brezinski, Journal of Approximation Theory, February, 2010) From the Back Cover 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, namely 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!

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This book has a wealth of practical information about using continued fractions to compute functions and constants. Great attention is paid to comparing several continued fraction formulations and series formulations for accuracy for each function.

⭐”In the words of Etta Jones, “At Last!” I have written before about the superiority of continued fractions. Most are familiar with infinite series in the form of sums (symbolized with a capital SIGMA) and products(PI). Fewer are familiar with nested series (LAMBDA); and fewer still with continued fractions (K, for Kettenbruch, lit. “chain fraction” in German).As an example of the four forms, consider Euler’s constant, e, represented four ways –sum: 1 + 1 + 1/2! + 1/3! + …product: (1)(2)(5/4)(16/15)…nested: 1+[1+(1/2)[1+(1/3)[1+…Continued fraction: 1/[1-1/[2-1/[3-2/[4-…In the above example, all series generate the same sequence of approximations for e, namely: 1, 2, 5/2, 8/3, 65/24, 326/120…. However, that is not to say that other sequences do not exist or are not more efficient. Indeed, continued fractions with unit numerators generally represent the best rational approximations. For example, e = 2+1/[1+1/[2+1/[1+1/[1+1/[4+1/[1+1/[1+1/[6+… — a regular sequence having numerators of unity and denominators of even numbers interspersed with two unit denominators — generates the following sequence of rational approximations: 2, 3, 8/3, 11/4, 19/7, 87/32, 106/39…. This latter sequence is actually a series of best rational approximations for e. Don’t believe me? Try e ~ 2721/1001, the eleventh in the sequence; the associated error is in the seventh decimal place.For a compendium of sums and nested series, see Jan J. Tuma’s “Engineering Mathematics Handbook,” an exquisite handbook that gives them for virtually every important mathematical function. However, continued fractional representations are much harder to come by in accumulated form. That is, until now.”Handbook of Continued Continued Fractions for Special Functions” is the uber reference for continued fractions. The book is organized into three parts and 20 chapters: Part 1, Basic Theory; Part 2, Numerics; and Part 3, Special Functions. Part 1 is indispensable in forming a coherent framework of the whole; my only criticism of the book occurs here — examples for each technique would have aided the reader in understanding them. Notwithstanding, Part 1 cumulates virtually all known methods for manipulating continued fractions; Part 2 will be of interest to those constructing numerical algorithms; but Part 3 was my very favorite because continued fractional representations of nearly every mathematical constant or function are given — often in several different ways. These include constants such as pi, e, ln(2), root(2), the golden ratio, Euler’s constant: gamma, Catalan’s constant, and more. Functions include circular and hyperbolic functions with their inverses; the gamma, incomplete gamma, and polygamma functions; the error function and Fresnel integrals; and hypergeometric, Bessel, and probability functions.As with most technical books, the price tag of ~$100 may be a bit rich for some. However, the reference is absolutely unique and indispensable for those who want to learn and work with continued fractions. Although written for the neophyte, it does require applied effort on the part of the reader. However, the reward is well worth it. Cuyt et al have surely contributed to the art with their publication.

⭐Libro muy completo .pero su precio no es bajo

⭐

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