By William B. Jones, W. J. Thron

ISBN-10: 051175955X

ISBN-13: 9780511759550

ISBN-10: 0521101522

ISBN-13: 9780521101523

ISBN-10: 0521302315

ISBN-13: 9780521302319

This is often an exposition of the analytic thought of endured fractions within the complicated area with emphasis on purposes and computational tools.

**Read or Download Continued Fractions: Analytic Theory and Applications (Encyclopedia of Mathematics and its Applications (No. 11)) PDF**

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**Extra info for Continued Fractions: Analytic Theory and Applications (Encyclopedia of Mathematics and its Applications (No. 11))**

**Sample text**

No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of the publisher. 1 D. Schwenter in 1625 and C. Huygens (1629-1695) in a posthumous publication considered the approximants of finite regular continued fractions as a means of expressing large fractions approximately in terms of fractions involving smaller numbers. Thus Schwenter (but in very awkward notation) had iZZ-I }L 1 1.

E. 8). Hurwitz also showed that V5 is the smallest value for which this result is true for all x. A measure-theoretic flavor was added to the theory by Borel [1909] and F. Bernstein (1878-1956) [1912], who proved that for almost all x, 0

7). Hence there exists a non-zero constant r m+1 such that a* =r r a ^ ^* =r b . 3). 3). 12) and S:(rnw) = Sn(w), /i = 0 , l , 2 , . . 4). 13). 3) is called an equivalence transformation. In subsequent chapters it is shown that certain properties of continued fractions are invariant under equivalence transformations. An equivalence transformation can be expressed by writing for infinite continued fractions and for finite continued fractions. Before looking at examples of equivalence transformations, we consider briefly the question of what sequences can form the approximants of continued fractions.

### Continued Fractions: Analytic Theory and Applications (Encyclopedia of Mathematics and its Applications (No. 11)) by William B. Jones, W. J. Thron

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