zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Continued fractions with applications. (English) Zbl 0782.40001
Studies in Computational Mathematics. 3. Amsterdam: North-Holland. xvi, 606 p. Dfl. 275.00; $ 157.00 (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).

40-02Research monographs (sequences, series, summability)
40A15Convergence and divergence of continued fractions
11-02Research monographs (number theory)
11A55Continued fractions (number-theoretic results)
30-02Research monographs (functions of one complex variable)
30B70Continued fractions (function-theoretic results)