Continued fractions with applications.

*(English)*Zbl 0782.40001
Studies in Computational Mathematics. 3. Amsterdam: North-Holland. xvi, 606 p. (1992).

Every mathematician sooner or later meets the concept of a continued fraction. It should actually be part of the curriculum giving one of the fine opportunities to study a concept that can not be boxed in one sub- discipline! The interesting interplay between recurrence relations, difference equations, orthogonal polynomials, moment problems, approximation theory, geometry of zeroes etc. has led and is still leading to beautiful mathematics.

This book is not meant to be ‘the’ new text on continued fractions, making older works obsolete (as the authors state), but serves as a gentle introduction into the subject, urging people who are interested to pursue their endeavors by consulting the famous books by O. Perron [Die Lehre von den Kettenbrüchen; Teubner, Stuttgart, Band I (1954; Zbl 0056.059); Band II (1957; Zbl 0077.066)], H. S. Wall [Analytic theory of continued fractions; van Nostrand company, New York (1948; Zbl 0035.036)], A. N. Khovanskij [The application of continued fractions and their generalizations to problems in approximation theory; P. Noordhoff, Groningen (1963; Zbl 0106.272)] and W. B. Jones and W. J. Thron [Continued fractions, analytic theory and applications; Addison-Wesley (1980; Zbl 0445.30003), now distributed by Cambridge University Press, New York (1984; Zbl 0603.30009)] and journals and proceedings covering the ongoing research.

The book is not cheap but it is a must for every library. It can not only be used to get people interested in the subject itself, but it will also show how much mathematics can resemble a great experimental science where calculations and new theoretical insights go hand in hand to further a subject that easily crosses the sometimes unrealistic divisions between a lot of mathematical subdisciplines (including ‘pure’ and ‘applied’) and between mathematics and disciplines like electrotechnical engineering (design of filters, stability etc.)

The book consists of 12 chapters, an appendix (continued fractions expansions for elementary functions, hypergeometric functions, basic hypergeometric functions) and a subject index. Each of the chapters has its own set of problems and nearly all of the chapters are followed by remarks concerning ongoing research. The titles of the chapters speak for themselves: 1. Introductory examples (54 pages). 2. More basics (38 pages). 3. Convergence criteria (95 pages). 4. Continued fractions and three-term recurrence relations (54 pages). 5. Correspondence of continued fractions (51 pages). 6. Hypergeometric functions (40 pages). 7. Moments and orthogonality (36 pages). 8. Padé approximants (30 pages). 9. Some applications in Number Theory (44 pages). 10. Zero-free regions (40 pages). 11. Digital filters and continued fractions (40 pages). 12. Applications to some differential equations (38 pages).

This book is not meant to be ‘the’ new text on continued fractions, making older works obsolete (as the authors state), but serves as a gentle introduction into the subject, urging people who are interested to pursue their endeavors by consulting the famous books by O. Perron [Die Lehre von den Kettenbrüchen; Teubner, Stuttgart, Band I (1954; Zbl 0056.059); Band II (1957; Zbl 0077.066)], H. S. Wall [Analytic theory of continued fractions; van Nostrand company, New York (1948; Zbl 0035.036)], A. N. Khovanskij [The application of continued fractions and their generalizations to problems in approximation theory; P. Noordhoff, Groningen (1963; Zbl 0106.272)] and W. B. Jones and W. J. Thron [Continued fractions, analytic theory and applications; Addison-Wesley (1980; Zbl 0445.30003), now distributed by Cambridge University Press, New York (1984; Zbl 0603.30009)] and journals and proceedings covering the ongoing research.

The book is not cheap but it is a must for every library. It can not only be used to get people interested in the subject itself, but it will also show how much mathematics can resemble a great experimental science where calculations and new theoretical insights go hand in hand to further a subject that easily crosses the sometimes unrealistic divisions between a lot of mathematical subdisciplines (including ‘pure’ and ‘applied’) and between mathematics and disciplines like electrotechnical engineering (design of filters, stability etc.)

The book consists of 12 chapters, an appendix (continued fractions expansions for elementary functions, hypergeometric functions, basic hypergeometric functions) and a subject index. Each of the chapters has its own set of problems and nearly all of the chapters are followed by remarks concerning ongoing research. The titles of the chapters speak for themselves: 1. Introductory examples (54 pages). 2. More basics (38 pages). 3. Convergence criteria (95 pages). 4. Continued fractions and three-term recurrence relations (54 pages). 5. Correspondence of continued fractions (51 pages). 6. Hypergeometric functions (40 pages). 7. Moments and orthogonality (36 pages). 8. Padé approximants (30 pages). 9. Some applications in Number Theory (44 pages). 10. Zero-free regions (40 pages). 11. Digital filters and continued fractions (40 pages). 12. Applications to some differential equations (38 pages).

Reviewer: M.G.de Bruin (Haarlem)

##### MSC:

40-02 | Research exposition (monographs, survey articles) pertaining to sequences, series, summability |

40A15 | Convergence and divergence of continued fractions |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11A55 | Continued fractions |

30-02 | Research exposition (monographs, survey articles) pertaining to functions of a complex variable |

30B70 | Continued fractions; complex-analytic aspects |