By S. Clement Cooper (auth.), Lisa Jacobsen (eds.)

**Contents: **S.C. Cooper: -Fraction options to Riccati Equations.- R.M. Hovstad: Irrational persevered Fractions.- L. Jacobsen, W.J. Thron, H. Waadeland: Julius Worpitzky, his Contributions to the Analytic thought of persisted Fractions and his Times.- W.B. Jones, N.J. Wyshinski: optimistic T-Fraction Expansions for a kinfolk of exact Functions.- J.H. McCabe: On persisted Fractions linked to Polynomial sort Padé Approximants, with an Application.- O. Njastad: Multipoint Padé Approximants and similar persevered Fractions.- O. Njastad: A Survey of a few effects on Separate Convergence of endured Fractions.- O. Njastad, H. Waadeland: a few comments on Nearness difficulties for persevered Fraction Expansions.- W.J. Thron: persevered Fraction Identities Derived from the Invariance of the Crossratio less than Linear Fractional Transformations.- H. Waadeland: Boundary types of Worpitzky's Theorem and of Parabola Theorems.

**Read or Download Analytic Theory of Continued Fractions III: Proceedings of a Seminar-Workshop, held in Redstone, USA, June 26–July 5, 1988 PDF**

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**Extra resources for Analytic Theory of Continued Fractions III: Proceedings of a Seminar-Workshop, held in Redstone, USA, June 26–July 5, 1988**

**Example text**

Worpitzky was familiar with Kliigel's WSrterbuch. He referred to it repeatedly in his article in Crelle's Journal in 1883. So it is not surprising to find in both of Worpitzky's articles on continued fractions (1862 and 1865) as a necessary and sufficient condition for the convergence of a continued fraction with approximants fn the condition fn - f,-I "-~ 0. Standards of rigor in mathematical arguments were increasing throughout the 19th century. But again this spread was not uniform. Grunert in particular may not have been much influenced by this trend.

1a). 1d). 1b) follows from integration by parts with u = t ~, k # 0 a n d v = n ( t -/3)~. 1b). Remark. 1e) is used to compute ck for negative k <_ - 2 . T h e o r e m 2 . 2 b Let/3 < 0, 0 < a < [BI and a positive integer n be given. Let ~n(t) := n(t - / 3 ) ~ , - a - / 3 < t < - ~ . aa 1 {-(-~)-~-'~nO - ,~) + 2(-~)- ~-' Z j=o ,t-tl T ~ . { 2 j + 1 \ 11(~) ~ ~ ~s s,nt - - - - ~ ) , , / j ° , j=o i f n is odd. 2a). 2d) is obtalned. , k # o, and v = - n ( - , O - ~)~. Remark. 2c) is used to compute ct` for negative k < - 2 .

Szgsz referred to Worpitzky at least once more. T h a t was in an article presented to the "Sitzungsberichte der Miinchner Akademie" by Pringsheim in 1919. The first reference in a text on continued fractions is by Wall (a student of Van Vleck) in his "Analytic Theory of Continued Fractions" in 1948. He calls the result "Worpitzky's Theorem" (Chapter III, §10) and introduces it with the comment: In what appears to be the earliest published paper treating of the convergence of continued fractions with complex elements, Worpitzky showed t h a t 1/(1 + K 2 (a J 1 ) ) converges if the partial numerators a2, a3, a 4 , ' ' " all have moduli tess than 1/4.