Download e-book for kindle: Continued Fractions: Analytic Theory and Applications by William B. Jones, W. J. Thron

By William B. Jones, W. J. Thron

This can be an exposition of the analytic conception of persisted fractions within the advanced area with emphasis on purposes and computational tools.

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

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

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1 History 5 2. There exists a constant y, independent of x, such that for almost all x lim n [Khintchine, 1936]. 3 Analytic Theory Euler made important contributions to the analytic theory. He gave continued-fraction expansions (always without convergence considerations) of integrals and power series, including divergent ones. He also showed how Brouncker's expression for 4/V could be derived from either Wallis's product formula or the Gregory-Leibniz alternating series for TT/4. Another of Euler's contributions was a solution of the Riccati differential equation in terms of continued fractions.

3). J = oo. Further additions to convergence theory, in particular the limit-periodic continued fractions, were made by Pringsheim in Miinchen, his student O. Perron (1880-1973), who also became a professor in Miinchen, and O. Szasz (1884-1952). Szasz spent a year in Miinchen before moving on to Frankfurt (where he became a colleague of Hellinger). Later he came to Cincinnati. 2 Overview of Contents 11 outweighed by his scholarly work on continued fractions. The three editions of his book "Die Lehre von den Kettenbriichen" [1913; 1929; 1954, 1957a] not only kept interest in the subject alive but also provided a model of expository writing.

3d) Dn = cnDn_x + dnBn_x, n= 1 , 2 , 3 , . . 3. t! 3). 4a) « = 0,1,2,.... 4b) n and AnDn-BnCn¥=0, Moreover AnDn-BnCn = (-iyJl^akdk-bkck), /i = 0 , l , 2 , . . 2 Proof. 3) by straightforward computation. 3). 4b) follows. 5). 1. 3 suppose that co=l,

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