×

Inequalities for the perimeter of an ellipse. (English) Zbl 0985.26009

The authors describe a method to study whether an algebraic approximation to the perimeter of an ellipse is from above or below. By the representation of the perimeter in terms of hypergeometric functions the problem boils down to establishing the sign of the error \[ E(x)=F(1/2, -1/2; 1; x)-A(x) , \] where \(A(x)\) is an algebraic function (depending on the approximation chosen) of the parameter \(x \in (0,1)\) related to the eccentricity of the ellipse. This problem can be tackled analyzing the sign of a series whose entries are all \(>0\) starting from a sufficiently large index. Thus, the question is reduced to the sign of a polynomial given by the sum of a finite number of terms of the series. In the situation described its coefficients are integers, and we can apply a Sturm sequence argument with the aid of a computer algebra system performing integer arithmetics.
In this way, the authors show that several classical formulas approximate the elliptical perimeter from below, proving in particular a conjecture by Vuorinen on a Muir’s formula.

MSC:

26D07 Inequalities involving other types of functions
33C05 Classical hypergeometric functions, \({}_2F_1\)
33C75 Elliptic integrals as hypergeometric functions
41A30 Approximation by other special function classes
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Almkvist, G.; Berndt, B., Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, π, and the Ladies Diary, Amer. Math. Monthly, 95, 585-608 (1988) · Zbl 0665.26007
[2] Berndt, B., Ramanujan’s Notebooks, Part III (1985), Springer-Verlag: Springer-Verlag New York/Berlin · Zbl 0555.10001
[3] Barnard, R. W.; Pearce, K.; Richards, K., An inequality involving the generalized hypergeometric function and the arc length of an ellipse, SIAM J. Math. Anal., 32, 403-419 (2000)
[4] Devlin, K., The logical structure of computer-aided mathematical reasoning, Amer. Math. Monthly, 104, 632-646 (1997) · Zbl 0891.00001
[5] Henrici, P., Applied and Computational Complex Analysis (1974), Wiley: Wiley New York
[6] Vuorinen, M., Hypergeometric functions in geometric function theory, (Srinivasa Roa, K.; Jagannthan, P.; Van der Jeugy, G., Proceedings of Special Functions and Differential Equations (1998), Allied Publishers) · Zbl 0948.30024
[7] Young, D.; Gregory, R., A Survey of Numerical Mathematics (1973), Dover: Dover New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.