How is the period of a simple pendulum growing with increasing amplitude?

References

[1] Abramowitz, M.—Stegun, I. A.: Handbook of Mathematical Functions, 9th edn., Dover Publications, New York, 1974.Search in Google Scholar

[2] Adlaj, S.: An eloquent formula for the perimeter of an ellipse, Notices of the AMS 59 (2012), 1094–1099.10.1090/noti879Search in Google Scholar

[3] Almkvist, G.—Berndt, B.: Gauss, Landen, Ramanujan, the arithmetic–geometric mean, ellipses, π, and the Ladies Diary, Amer. Math. Monthly 95 (1988), 585–608.10.1080/00029890.1988.11972055Search in Google Scholar

[4] Amrani, D.—Paradis, P.—Beaudin, M.: Approximation expressions for the large-angle period of a simple pendulum revisited, Rev. Mex. Fís. E 54 (2008), 59–64.Search in Google Scholar

[5] Beléndez, A. et al.: Analytical approximations for the period of a nonlinear pendulum, Eur. J. Phys. 27 (2006), 539–551.10.1088/0143-0807/27/3/008Search in Google Scholar

[6] Beléndez, A. et al.: Approximation for the large-angle simple pendulum period, Eur. J. Phys. 30 (2009), 25–28.10.1088/0143-0807/30/2/L03Search in Google Scholar

[7] Beléndez, A. et al.: Higher accurate approximate solutions for the simple pendulum in terms of elementary functions, Eur. J. Phys. 31 (2010), 65–70.10.1088/0143-0807/31/3/L04Search in Google Scholar

[8] Beléndez, A. et al.: Approximate solutions for the nonlinear pendulum equation using a rational harmonic representation, Comput. Math. Appl. 64 (2012), 1602–1611.10.1016/j.camwa.2012.01.007Search in Google Scholar

[9] Borghi, R.: Simple pendulum dynamics: revisiting the Fourier-basedapproach to the solution, arXiv:1303.5023v1[physics.class-ph].Search in Google Scholar

[10] Carvalhaes, C. G.—Suppes, P.: Approximation for the period of the simple pendulum based on the arithmetic–geometric mean, Am. J. Phys. 76 (2008), 1150–1154.10.1119/1.2968864Search in Google Scholar

[11] Chen, C.-P.—Qi, F.: Best upper and lower bounds in Wallis’ inequality, J. Indones. Math. Soc. 11 (2005), 137–141.Search in Google Scholar

[12] Chen, C.-P.—Qi, F.: The best bounds in Wallis’ inequality, Proc. Amer. Math. Soc. 133 (2005), 397–401.10.1090/S0002-9939-04-07499-4Search in Google Scholar

[13] Chen, C.-P.—Qi, F.: Completely monotonic function associated with the gamma functions and proof of Wallis’ inequality, Tamkang J. Math. 36 (2005), 303–307.10.5556/j.tkjm.36.2005.101Search in Google Scholar

[14] Cristea, V. G.: A direct approach for proving Wallis’ ratio estimates and an improvement of Zhang-Xu-Situ inequality, Stud. Univ. Babeş-Bolyai Math. 60 (2015), 201–209.Search in Google Scholar

[15] Dadfar, M. B.–Geer, J. F.: Power series solution to a simple pendulum with oscillating support, SIAM J. Appl. Math. 47 (1987), 737–750.10.1137/0147051Search in Google Scholar

[16] Deng, J.-E.—Ban, T.—Chen, C.-P.: Sharp inequalities and asymptotic expansion associated with the Wallis sequence, J. Inequal. Appl. (2015), 2015:186.10.1186/s13660-015-0699-zSearch in Google Scholar

[17] Dumitrescu, S.: Estimates for the ratio of gamma functions using higher order roots, Stud. Univ. Babeş-Bolyai Math. 60 (2015), 173–181.Search in Google Scholar

[18] Guo, S.—Xu, J.-G.—Qi, F.: Some exact constants for the approximation of the quantity in the Wallis’ formula, J. Inequal. Appl. (2013), 2013:67.10.1186/1029-242X-2013-67Search in Google Scholar

[19] Guo, S. et al.: A sharp two-sided inequality for bounding the Wallis ratio, J. Inequal. Appl. (2015), 2015:43.10.1186/s13660-015-0560-4Search in Google Scholar

[20] Guo, B.-N.—Qi, F.: On the Wallis formula, Internat. J. Anal. Appl. 8 (2015), 30–38.Search in Google Scholar

[21] Johannessen, K.: An anharmonic solution to the equation of motion for the simple pendulum, Eur. J. Phys. 32 (2011), 407–417.10.1088/0143-0807/32/2/014Search in Google Scholar

[22] Laforgia, A.—Natalini, P.: On the asymptotic expansion of a ratio of gamma functions, J. Math. Anal. Appl. 389 (2012), 833-837.10.1016/j.jmaa.2011.12.025Search in Google Scholar

[23] Lampret, V.: Wallis’ sequence estimated accurately using an alternating series, J. Number. Theory 172 (2017), 256-269.10.1016/j.jnt.2016.08.014Search in Google Scholar

[24] Lampret, V.: A simple asymptotic estimate of Wallis’ ratio using Stirling’s factorial formula, Bull. Malays. Math. Sci. Soc. 42 (2019), 3213–3221.10.1007/s40840-018-0654-5Search in Google Scholar

[25] Landau, L. D.—Lifshitz, E. M.: Course of Theoretical Physics: Mechanics, 3ed., Butterworth-Heinemann, 1986.Search in Google Scholar

[26] Lima, F. M. S.—Arun, P.: An accurate formula for the period of a simple pendulum oscillating beyond the small angle regime, Am. J. Phys. 74 (2006), 892–895.10.1119/1.2215616Search in Google Scholar

[27] Lima, F. M. S.: Simple ‘log formulae’ for pendulum motion valid for any amplitude, Eur. J. Phys. 29 (2008), 1091–1098.10.1088/0143-0807/29/5/021Search in Google Scholar

[28] Maclaurin, C. A.: A Treatise of Fluxions. In Two Books, Vol.2, T. W. and T. Ruddimans, Edinburgh, 1742.Search in Google Scholar

[29] Mortici, C.: Sharp inequalities and complete monotonicity for the Wallis ratio, Bull. Belg. Math. Math. Soc. Simon Stevin 17 (2010), 929–936.10.36045/bbms/1292334067Search in Google Scholar

[30] Mortici, C.: New approximation formulas for evaluating the ratio of gamma functions, Math. Comput. Modelling 52 (2010), 425–433.10.1016/j.mcm.2010.03.013Search in Google Scholar

[31] Mortici, C.: A new method for establishing and proving new bounds for the Wallis ratio, Math. Inequal. Appl. 13 (2010), 803–815.10.7153/mia-13-58Search in Google Scholar

[32] Mortici, C.: Completely monotone functions and the Wallis ratio, Appl. Math. Lett. 25 (2012), 717–722.10.1016/j.aml.2011.10.008Search in Google Scholar

[33] Mortici, C.—Cristea, V. G.: Estimates for Wallis’ ratio and related functions, Indian J. Pure Appl. Math. 47 (2016), 437–447.10.1007/s13226-016-0176-5Search in Google Scholar

[34] Parwani, R. R.: An approximate expression for the large angle period of a simple pendulum, Eur. J. Phys. 25 (2004), 37–39.10.1088/0143-0807/25/1/006Search in Google Scholar

[35] Qi, F.—Mortici, C.: Some best approximation formulas and the inequalities for the Wallis ratio, Appl. Math. Comput. 253 (2015), 363–368.10.1016/j.amc.2014.12.039Search in Google Scholar

[36] Qi, F.: An improper integral, the beta function, the Wallis ratio, and the Catalan numbers, Prob. Anal. Issues Anal. 7 (2018), 104–115.10.15393/j3.art.2018.4370Search in Google Scholar

[37] Siboni, S.: Superlinearly convergent homogeneous maps and period of the pendulum, Am. J. Phys. 75 (2007), 368–373.10.1119/1.2426350Search in Google Scholar

[38] Sun, J.-S.—Qu, C.-M.: Alternative proof of the best bounds of Wallis’ inequality, Commun. Math. Anal. 2 (2007), 23–27.Search in Google Scholar

[39] Villarino, M. B.: The AGM simple pendulum, arXiv:1202.2782v2, (2014).Search in Google Scholar

[40] Wolfram, S.: Mathematica, Version 7.0, Wolfram Research, Inc., 1988–2009.Search in Google Scholar

[41] Zhang, X.-M.—Xu, T. Q.—Situ, L. B.: Geometric convexity of a function involving gamma function and application to inequality theory, J. Inequal. Pure Appl. Math. 8 (2007), Art. 17.Search in Google Scholar