The perimeter of a flattened ellipse can be estimated accurately even from Maclaurin‘s series
-
Vito Lampret
vito.lampret@guest.arnes.si
Downloads
DOI:
https://doi.org/10.4067/S0719-06462019000200051Abstract
For the perimeter \(P(a,b)\) of an ellipse with the semi-axes \(a\ge b\ge 0\) a sequence \(Q_n(a,b)\) is constructed such that the relative error of the approximation \(P(a,b)\approx Q_n(a,b)\) satisfies the following inequalities
\(0\le -\frac{P(a,b)-Q_n(a,b)}{P(a,b)}\le\frac{(1-q^2)^{n+1}}{(2n+1)^2}\)
\(\le \frac{1}{(2n+1)^2}\,e^{-q^2(n+1)},\)
true for \(n\in{\mathbb N}\) and \(q=\frac{b}{a}\in[0,1]\).
Keywords
[1] S. Adlaj, An eloquent formula for the perimeter of an ellipse, Notices of the AMS 59 (2012), no. 8, 1094–1099.
[2] G. Almkvist and B. Berndt, Gauss, Landen, Ramanujan, the arithmetic–geometric mean, ellipses, Ï€, and the Ladies Diary, Amer. Math. Monthly 95 (1988), 585–608.
[3] B. W. Barnard, K. Pearce and L. Schovanec, Inequalities for the perimeter of an Ellipse, J. Math. Anal. Appl. 260 (2001), 295–306.
[4] C.-P. Chen and F. Qi, Best upper and lower bounds in Wallis‘ inequality, Journal of the Indonesian Mathematical Society 11 (2005), no. 2, 137–141.
[5] C.-P. Chen and F. Qi, The best bounds in Wallis‘ inequality, Proc. Amer. Math. Soc. 133(2005), 397–401.
[6] C.-P. Chen and F. Qi, Completely monotonic function associated with the gamma functions and proof of Wallis‘ inequality, Tamkang Journal of Mathematics 36 (2005), no. 4, 303–307.
[7] V. G. Cristea, A direct approach for proving Wallis‘ ratio estimates and an improvement of Zhang-Xu-Situ inequality, Studia Univ. Babes,-Bolyai Math. 60 (2015), 201–209.
[8] J.-E. Deng, T. Ban and C.-P. Chen;Sharp inequalities and asymptotic expansion associated with the Wallis sequence, J. Inequal. Appl., (2015), 2015:186.
[9] S. Dumitrescu, Estimates for the ratio of gamma functions using higher order roots, Studia Univ. Babes,-Bolyai Math. 60 (2015), 173–181.
[10] S. Guo, J.-G. Xu and F. Qi,Some exact constants for the approximation of the quantity in the Wallis‘ formula, J. Inequal. Appl., (2013), 2013:67.
[11] S. Guo, Q. Feng, Y.-Q. Bi and Q.-M. Luo, A sharp two-sided inequality for bounding the Wallis ratio, J. Inequal. Appl., (2015), 2015:43.
[12] B.-N. Guo and Feng Qi, On the Wallis formula, International Journal of Analysis and Appli- cations 8 (2015), no. 1, 30–38.
[13] J. Ivory, A new series for the rectification of the ellipsis; together with some observations on the evolution of the formula ô°†a2 + b2 − 2ab cos φô°‡n, Trans. Royal Soc. Edinburgh 4 (1796), 177–190.
[14] A. Laforgia and P. Natalini, On the asymptotic expansion of a ratio of gamma functions, J. Math. Anal. Appl. 389 (2012), 833-837.
[15] V. Lampret, The Euler-Maclaurin and Taylor Formulas: Twin, Elementary Derivations, Math. Mag. 74 (2001), No. 2, pp. 109–122.
[16] V. Lampret, Wallis‘ Sequence Estimated Accurately Using an Alternating Series, J. Number. Theory. 172 (2017), 256-269.
[17] V. Lampret, A Simple Asymptotic Estimate of Wallis Ratio Using Stirlings Factorial Formula, Bull. Malays. Math. Sci. Soc. (2018), doi.org/10.1007/s40840-018-0654-5.
[18] C. E. Linderholm and A. C. Segal, An Overlooked Series for the Elliptic Perimeter, Mathe- matics Magazine, 68(1995)3, 216–220.
[19] C. Mortici, Sharp inequalities and complete monotonicity for the Wallis ratio, Bull. Belg. Math. Math. Soc. Simon Stevin, 17 (2010), pp. 929–936.
[20] C. Mortici, New approximation formulas for evaluating the ratio of gamma functions, Math. Comput. Modelling 52 (2010), pp. 425–433.
[21] C. Mortici, A new method for establishing and proving new bounds for the Wallis ratio, Math. Inequal. Appl. 13 (2010), 803–815.
[22] C. Mortici, Completely monotone functions and the Wallis ratio, Appl. Math. Lett. 25 (2012), 717–722.
[23] C. Mortici and V. G. Cristea, Estimates for Wallis‘ ratio and related functions, Indian J. Pure Appl. Math. 47 (2016), 437–447.
[24] C. A. Maclaurin, A treatise of fluctions in two books, Vol.2, T. W. and T. Ruddimans, Edin- burgh 1742.
[25] F. Qi and C. Mortici, Some best approximation formulas and the inequalities for the Wallis ratio, Appl. Math. Comput. 253 (2015), 363–368.
[26] , F. Qi, An improper integral, the beta function, the Wallis ratio, and the Catalan numbers, Problemy Analiza–Issues of Analysis 7 (25) (2018), no. 1, 104–115.
[27] J.-S. Sun and C.-M. Qu, Alternative proof of the best bounds of Wallis‘ inequality, Commun. Math. Anal. 2 (2007), 23–27.
[28] M. B. Villarino, A Direct Proof of Landen‘s Transformation, arXiv:math/0507108v1 [math.CA].
[29] M. B. Villarino, Ramanujan‘s inverse elliptic arc approximation, Ramanujan J., 34 (2014), no. 2, 157–161.
[30] S. Wolfram, Mathematica, Version 7.0, Wolfram Research, Inc., 1988–2009.
[31] X.-M. Zhang, T. Q. Xu and L. B. Situ Geometric convexity of a function involving gamma function and application to inequality theory, J. Inequal. Pure Appl. Math. 8 (2007) 1, art. 17, 9 p.
[2] G. Almkvist and B. Berndt, Gauss, Landen, Ramanujan, the arithmetic–geometric mean, ellipses, Ï€, and the Ladies Diary, Amer. Math. Monthly 95 (1988), 585–608.
[3] B. W. Barnard, K. Pearce and L. Schovanec, Inequalities for the perimeter of an Ellipse, J. Math. Anal. Appl. 260 (2001), 295–306.
[4] C.-P. Chen and F. Qi, Best upper and lower bounds in Wallis‘ inequality, Journal of the Indonesian Mathematical Society 11 (2005), no. 2, 137–141.
[5] C.-P. Chen and F. Qi, The best bounds in Wallis‘ inequality, Proc. Amer. Math. Soc. 133(2005), 397–401.
[6] C.-P. Chen and F. Qi, Completely monotonic function associated with the gamma functions and proof of Wallis‘ inequality, Tamkang Journal of Mathematics 36 (2005), no. 4, 303–307.
[7] V. G. Cristea, A direct approach for proving Wallis‘ ratio estimates and an improvement of Zhang-Xu-Situ inequality, Studia Univ. Babes,-Bolyai Math. 60 (2015), 201–209.
[8] J.-E. Deng, T. Ban and C.-P. Chen;Sharp inequalities and asymptotic expansion associated with the Wallis sequence, J. Inequal. Appl., (2015), 2015:186.
[9] S. Dumitrescu, Estimates for the ratio of gamma functions using higher order roots, Studia Univ. Babes,-Bolyai Math. 60 (2015), 173–181.
[10] S. Guo, J.-G. Xu and F. Qi,Some exact constants for the approximation of the quantity in the Wallis‘ formula, J. Inequal. Appl., (2013), 2013:67.
[11] S. Guo, Q. Feng, Y.-Q. Bi and Q.-M. Luo, A sharp two-sided inequality for bounding the Wallis ratio, J. Inequal. Appl., (2015), 2015:43.
[12] B.-N. Guo and Feng Qi, On the Wallis formula, International Journal of Analysis and Appli- cations 8 (2015), no. 1, 30–38.
[13] J. Ivory, A new series for the rectification of the ellipsis; together with some observations on the evolution of the formula ô°†a2 + b2 − 2ab cos φô°‡n, Trans. Royal Soc. Edinburgh 4 (1796), 177–190.
[14] A. Laforgia and P. Natalini, On the asymptotic expansion of a ratio of gamma functions, J. Math. Anal. Appl. 389 (2012), 833-837.
[15] V. Lampret, The Euler-Maclaurin and Taylor Formulas: Twin, Elementary Derivations, Math. Mag. 74 (2001), No. 2, pp. 109–122.
[16] V. Lampret, Wallis‘ Sequence Estimated Accurately Using an Alternating Series, J. Number. Theory. 172 (2017), 256-269.
[17] V. Lampret, A Simple Asymptotic Estimate of Wallis Ratio Using Stirlings Factorial Formula, Bull. Malays. Math. Sci. Soc. (2018), doi.org/10.1007/s40840-018-0654-5.
[18] C. E. Linderholm and A. C. Segal, An Overlooked Series for the Elliptic Perimeter, Mathe- matics Magazine, 68(1995)3, 216–220.
[19] C. Mortici, Sharp inequalities and complete monotonicity for the Wallis ratio, Bull. Belg. Math. Math. Soc. Simon Stevin, 17 (2010), pp. 929–936.
[20] C. Mortici, New approximation formulas for evaluating the ratio of gamma functions, Math. Comput. Modelling 52 (2010), pp. 425–433.
[21] C. Mortici, A new method for establishing and proving new bounds for the Wallis ratio, Math. Inequal. Appl. 13 (2010), 803–815.
[22] C. Mortici, Completely monotone functions and the Wallis ratio, Appl. Math. Lett. 25 (2012), 717–722.
[23] C. Mortici and V. G. Cristea, Estimates for Wallis‘ ratio and related functions, Indian J. Pure Appl. Math. 47 (2016), 437–447.
[24] C. A. Maclaurin, A treatise of fluctions in two books, Vol.2, T. W. and T. Ruddimans, Edin- burgh 1742.
[25] F. Qi and C. Mortici, Some best approximation formulas and the inequalities for the Wallis ratio, Appl. Math. Comput. 253 (2015), 363–368.
[26] , F. Qi, An improper integral, the beta function, the Wallis ratio, and the Catalan numbers, Problemy Analiza–Issues of Analysis 7 (25) (2018), no. 1, 104–115.
[27] J.-S. Sun and C.-M. Qu, Alternative proof of the best bounds of Wallis‘ inequality, Commun. Math. Anal. 2 (2007), 23–27.
[28] M. B. Villarino, A Direct Proof of Landen‘s Transformation, arXiv:math/0507108v1 [math.CA].
[29] M. B. Villarino, Ramanujan‘s inverse elliptic arc approximation, Ramanujan J., 34 (2014), no. 2, 157–161.
[30] S. Wolfram, Mathematica, Version 7.0, Wolfram Research, Inc., 1988–2009.
[31] X.-M. Zhang, T. Q. Xu and L. B. Situ Geometric convexity of a function involving gamma function and application to inequality theory, J. Inequal. Pure Appl. Math. 8 (2007) 1, art. 17, 9 p.
Most read articles by the same author(s)
- Vito Lampret, Basic asymptotic estimates for powers of Wallis‘ ratios , CUBO, A Mathematical Journal: Vol. 23 No. 3 (2021)
- Vito Lampret, Double asymptotic inequalities for the generalized Wallis ratio , CUBO, A Mathematical Journal: Vol. 26 No. 1 (2024)
Downloads
Download data is not yet available.
Published
2019-08-10
How to Cite
[1]
V. Lampret, “The perimeter of a flattened ellipse can be estimated accurately even from Maclaurin‘s series”, CUBO, vol. 21, no. 2, pp. 51–64, Aug. 2019.
Issue
Section
Articles
