By S. Clement Cooper (auth.), Lisa Jacobsen (eds.)
Contents: S.C. Cooper: -Fraction ideas to Riccati Equations.- R.M. Hovstad: Irrational persevered Fractions.- L. Jacobsen, W.J. Thron, H. Waadeland: Julius Worpitzky, his Contributions to the Analytic conception of persevered Fractions and his Times.- W.B. Jones, N.J. Wyshinski: optimistic T-Fraction Expansions for a relatives of detailed Functions.- J.H. McCabe: On persisted Fractions linked to Polynomial style Padé Approximants, with an Application.- O. Njastad: Multipoint Padé Approximants and comparable endured Fractions.- O. Njastad: A Survey of a few effects on Separate Convergence of endured Fractions.- O. Njastad, H. Waadeland: a few feedback on Nearness difficulties for persisted Fraction Expansions.- W.J. Thron: endured Fraction Identities Derived from the Invariance of the Crossratio below Linear Fractional Transformations.- H. Waadeland: Boundary types of Worpitzky's Theorem and of Parabola Theorems.
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Additional resources for Analytic Theory of Continued Fractions III: Proceedings of a Seminar-Workshop, held in Redstone, USA, June 26–July 5, 1988
This theorem was obtained 33 years earlier by Julius Worpitzky, then a teacher at the Friedrichs-Gymnasium in Berlin. Pringsheim returned numerous times to this result without ever acknowledging Worpitzky's priority, even though he must have known about Worpitzky well before 1925 when he included a long chapter on continued fractions with elements in the complex plane in his book "Zahlen- und Funktionenlehre". It is because of this, hard to understand, oversight, which was shared by many other mathematicians, t h a t we became interested in finding out who Worpitzky was, how he proved his theorem, and why nobody appears to have known about it before 1905.
It also has the distinction of being the first "convergence region" criterion. Since then many more results of this type, which in many respects is the most typical for convergence theorems for continued fractions, have been found. 2. E a r l y Life. Julius Daniel Theodor Worpitzky was born on May 10, 1835 in the small village of Carlsburg (now Karlsburg) near Greifswald in Pomerania. His father Johann Samuel (1804-59) was a school teacher in Cartsburg. His grandfather Johann Christian Worpitzky (1775-1857) was a damask weaver in Carlsburg as was one of his uncles (1802-56, committed suicide).
1 ak+nZ -- -- 1 E a c h Sn(k) is a r a t i o n a l function in z a n d h a s a T a y l o r series expansion vmO It w a s k n o w n before W o r p i t z k y t h a t p(k,n) is i n d e p e n d e n t of n for v = O, . . k), V = 0, . . , n + l . k)zv . v-O T h i s series, with k = 0, is t h e one referred to in the s t a t e m e n t of t h e t h e o r e m . ). Pn is a h o m o g e n e o u s polynomial in the v a r i a b l e s ak, . . , a k + n _ 1 of degree n with all coefficients positive. I <__ a, n ~ 1, converges for az -- Izl < 1 ..
Analytic Theory of Continued Fractions III: Proceedings of a Seminar-Workshop, held in Redstone, USA, June 26–July 5, 1988 by S. Clement Cooper (auth.), Lisa Jacobsen (eds.)