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Titolo:
A monotonicity property involving F-3(2) and comparisons of the classical approximations of elliptical arc length
Autore:
Barnard, RW; Pearce, K; Richards, KC;
Indirizzi:
Texas Tech Univ, Dept Math, Lubbock, TX 79409 USA Texas Tech Univ LubbockTX USA 79409 iv, Dept Math, Lubbock, TX 79409 USA SW Univ, Dept Math, Georgetown, TX 78626 USA SW Univ Georgetown TX USA 78626 Univ, Dept Math, Georgetown, TX 78626 USA
Titolo Testata:
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
fascicolo: 2, volume: 32, anno: 2000,
pagine: 403 - 419
SICI:
0036-1410(20000831)32:2<403:AMPIFA>2.0.ZU;2-4
Fonte:
ISI
Lingua:
ENG
Soggetto:
HYPERGEOMETRIC-FUNCTIONS; INEQUALITIES;
Keywords:
hypergeometric; approximations; elliptical arc length;
Tipo documento:
Article
Natura:
Periodico
Settore Disciplinare:
Physical, Chemical & Earth Sciences
Citazioni:
16
Recensione:
Indirizzi per estratti:
Indirizzo: Barnard, RW Texas Tech Univ, Dept Math, Lubbock, TX 79409 USA Texas Tech Univ Lubbock TX USA 79409 h, Lubbock, TX 79409 USA
Citazione:
R.W. Barnard et al., "A monotonicity property involving F-3(2) and comparisons of the classical approximations of elliptical arc length", SIAM J MATH, 32(2), 2000, pp. 403-419

Abstract

Conditions are determined under which F-3(2) (-n, a, b; a + b + 2, epsilon-n + 1; 1) is a monotone function of n satisfying ab.F-3(2)(-n, a, b; a + b+ 2, epsilon - n + 1; 1) greater than or equal to ab.F-2(1) (a, b; a + b 2; 1). Motivated by a conjecture of Vuorinen [ Proceedings of Special Functions and Differential Equations, K. S. Rao, R. Jagannathan, G. Vanden Berghe, J. Van der Jeugt, eds., Allied Publishers, New Delhi, 1998], the corollary that F-3(2) (-n, -1/2, -1/2; 1, epsilon - n + 1; 1) greater than or equal to 4/pi, for 1 > epsilon greater than or equal to 1/4 and n greater thanor equal to 2, is used to determine surprising hierarchical relationships among the 13 known historical approximations of the arc length of an ellipse. This complete list of inequalities compares the Maclaurin series coefficients of F-2(1) with the coefficients of each of the known approximations, for which maximum errors can then be established. These approximations range over four centuries from Kepler's in 1609 to Almkvist's in 1985 and include two from Ramanujan.

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Documento generato il 05/12/20 alle ore 01:41:56