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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!
List of contents
Basic Theory.- Basics.- Continued fraction representation of functions.- Convergence criteria.- Padé approximants.- Moment theory and orthogonal functions.- Numerics.- Continued fraction construction.- Truncation error bounds.- Continued fraction evaluation.- Special Functions.- On tables and graphs.- Mathematical constants.- Elementary functions.- Gamma function and related functions.- Error function and related integrals.- Exponential integrals and related functions.- Hypergeometric functions.- Confluent hypergeometric functions.- Bessel functions.- Probability functions.- Basic hypergeometric functions.
Summary
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!
Additional text
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)
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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)
