Research article

Concise high precision approximation for the complete elliptic integral of the first kind

  • Received: 13 June 2021 Accepted: 19 July 2021 Published: 28 July 2021
  • MSC : 33E05, 26E60

  • In this paper, we obtain a concise high-precision approximation for $ \mathcal{K}(r) $:

    $ \begin{equation*} \frac{2}{\pi }\mathcal{K}(r){\rm{ }}>{\rm{ }}\frac{22\left( r^{\prime }\right) ^{2}+84r^{\prime }+22}{7\left( r^{\prime }\right) ^{3}+57\left( r^{\prime }\right) ^{2}+57r^{\prime }+7}, \end{equation*} $

    which holds for all $ r\in (0, 1) $, where $ \mathcal{K}(r) $ is complete elliptic integral of the first kind and $ r^{\prime } = \sqrt{1-r^{2}} $.

    Citation: Ling Zhu. Concise high precision approximation for the complete elliptic integral of the first kind[J]. AIMS Mathematics, 2021, 6(10): 10881-10889. doi: 10.3934/math.2021632

    Related Papers:

  • In this paper, we obtain a concise high-precision approximation for $ \mathcal{K}(r) $:

    $ \begin{equation*} \frac{2}{\pi }\mathcal{K}(r){\rm{ }}>{\rm{ }}\frac{22\left( r^{\prime }\right) ^{2}+84r^{\prime }+22}{7\left( r^{\prime }\right) ^{3}+57\left( r^{\prime }\right) ^{2}+57r^{\prime }+7}, \end{equation*} $

    which holds for all $ r\in (0, 1) $, where $ \mathcal{K}(r) $ is complete elliptic integral of the first kind and $ r^{\prime } = \sqrt{1-r^{2}} $.



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    [1] F. Bowman, Introduction to Elliptic Function with Applications, Dover Publications, New York, 1961.
    [2] P. F. Byrd, M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, Springer-Verlag, New York, 1971.
    [3] G. D. Anderson, M. K. Vamanamurthy, M. Vuorinen, Conformal Invariants, Inequalities, and Quasiconformal Maps, John Wiley & Sons, New York, 1997.
    [4] G. D. Anderson, S. L. Qiu, M. Vuorinen, Precise estimates for differences of the Gaussian hypergeometric function, J. Math. Anal. Appl., 215 (1997), 212–234. doi: 10.1006/jmaa.1997.5641
    [5] S. Ponnusamy, M. Vuorinen, Univalence and convexity properties for Gaussian hypergeometric functions, Rocky Mountain J. Math., 31 (2001), 327–353.
    [6] Y. Q. Song, P. G. Zhou, Y. M. Chu, Inequalities for the Gaussian hypergeometric function, Sci. China Math., 57 (2014), 2369–2380. doi: 10.1007/s11425-014-4858-3
    [7] M. K. Wang, Y. M. Chu, Y. P. Jiang, Ramanujan's cubic transformation inequalities for zero-balanced hypergeometric functions, Rocky Mountain J. Math., 46 (2016), 679–691.
    [8] M. K. Wang, Y. M. Chu, Y. Q. Song, Asymptotical formulas for Gaussian and generalized zero-balanced hypergeometric functions, Appl. Math. Comput., 276 (2016), 44–60.
    [9] M. K. Wang, Y. M. Chu, Refinenemts of transformation inequalities for zero-balanced hypergeometric functions, Acta Math. Sci., 37 (2017), 607–622. doi: 10.1016/S0252-9602(17)30026-7
    [10] B. C. Carlson, M. Vuorinen, Inequality of the AGM and the logarithmic mean, SIAM. Rev., 33 (1991), 653–654.
    [11] M. K. Vamanamurthy, M. Vuorinen, Inequalities for means, J. Math. Anal. Appl., 183 (1994), 155–166.
    [12] S. L. Qiu, M. K. Vamanamurthy, Sharp estimates for complete elliptic integrals, SIAM. J. Math. Anal., 27 (1996), 823–834. doi: 10.1137/0527044
    [13] H. Alzer, Sharp inequalities for the complete elliptic integral of the first kind, Math. Proc. Cambridge Philos. Soc., 124 (1998), 309–314. doi: 10.1017/S0305004198002692
    [14] G. D. Anderson, M. K. Vamanamurthy, M. Vuorinen, Functional inequalities for hypergeometric functions and complete elliptic integrals, SIAM. J. Math. Anal., 23 (1992), 512–524. doi: 10.1137/0523025
    [15] H. Alzer, S. L. Qiu, Monotonicity theorem and inequalities for the complete elliptic integrals, J. Comput. Appl. Math., 172 (2004), 289–312. doi: 10.1016/j.cam.2004.02.009
    [16] Zh. H. Yang, Y. Q. Song, Y. M. Chu, Sharp bounds for the arithmetic-geometric mean, J. Inequal. Appl., 192 (2014), 13.
    [17] Z. H. Yang, W. M. Qian, Y. M. Chu, W. Zhang, On approximating the arithmetic-geometric mean and complete elliptic integral of the first kind, J. Math. Anal. Appl., 462 (2018), 1714–1726. doi: 10.1016/j.jmaa.2018.03.005
    [18] Z. H. Yang, J. F. Tian, Y. R. Zhu, A rational approximation for the complete elliptic integral of the first kind, Mathematics., 8 (2020), 635. doi: 10.3390/math8040635
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