Continued Fractions

Abstract

Aims and scope of the seriesThe series 'Atlantis Studies in Mathematics for Engineering and Science'(AMES) publishes high quality monographs in applied mathematics, computational mathematics, and statistics that have the potential to make a significant impact on the advancement of engineering and science on the one hand, and economics and commerce on the other. We welcome submission of book proposals and manuscripts from mathematical scientists worldwide who share our vision of mathematics as the engine of progress in the disciplines mentioned above.All books in this series are co-published with World Scientific.For more information on this series and our other book series, please visit our website at: For information on all Atlantis Press publications, visit our website at: www.atlantis-press.com. www.atlantis-press.com/publications/books © ATLANTIS PRESS / WORLD SCIENTIFIC AMSTERDAM -PARIS Continued Fractions CopyrightThis book, or any parts thereof, may not be reproduced for commercial purposes in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system known or to be invented, without prior permission from the Publisher. The interest in using continued fractions to approximate special functions has also grown since then. Such fractions are easy to program, they have impressive convergence properties, and their convergence is often easy to accelerate. They even have good and reliable truncation error bounds which makes it possible to control the accuracy of the approximation. The bounds are both of the a posteriori type which tells the accuracy of a done calculation, and of the a priori type which can be used to determine the number of terms needed for a wanted accuracy. This important aspect is treated in this first volume of the second edition, along with the basic theory.

Hausdorff dimension of sets of divergence arising from continued fractions

“…Usual statements of the Hillam-Thron Theorem, such as [10, Theorem 4.37], have essentially the same hypotheses as Theorem 4.1, but they have only this weaker conclusion that S n converges on D to a constant. There are variants on the Hillam-Thron Theorem (such as [11,Lemma 3.8]) in which the hypothesis s n (v) = u is weakened, and our observations on sets of divergence apply to most, if not all, of these alternative theorems.…”

“…The Seidel-Stern Theorem ([4, Theorem 1.8] or [11,Theorem 3.13]) states that, with our hypotheses, T n converges at 0 to a point p. Since T n (∞) = T n−1 (0) we also have that T n (∞) → p as n → ∞. Since T n converges to p at two distinct points we see from Lemma 3.1 that T n converges generally to p. Next, observe that t −1 n (−∞, 0) ⊆ (−∞, 0) for each n so that, by Lemma 3.2, the Julia set of T n is contained in [−∞, 0].…”

The Parabola Theorem on Continued Fractions

“…The value of K(a n | 1) -that is, the limit of the sequence T n (0) -is necessarily contained in H (and it cannot be 0 or ∞). In fact, it is known that every element in H \{0, ∞} is the value of some such continued fraction (see, for example, [15,Thm. 3.47]).…”

“…The theorem was extended by some of these authors in a number of subsequent papers including [14,21], and the statement of the theorem from [21] is recast in the books by Jones and Thron [11,Thm. 4.42] and Lorentzen and Waadeland [15,Thm. 3.43].…”

A Worpitzky boundary theorem for branched continued fractions of the special form

“…Such domains are indicated in the complex plane, that if elements a k , b k of a continued fraction belong to these domains then the continued fraction converges. At first convergence domains for continued fractions we can find in papers of Worpitzky (1865), Pringsheim (1899) and Van Vleck (1901) [8].…”