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A Natural Approximation to the Complete Elliptic Integral of the First Kind

Author

Listed:
  • Ling Zhu

    (Department of Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China)

Abstract

Let K ( r ) be the complete elliptic integral of the first kind. Then, the inequality 2 K ( r ) / π > tanh − 1 ( r ) / sin − 1 r holds for all r ∈ ( 0 , 1 ) . This conclusion does not match those in the existing literature.

Suggested Citation

  • Ling Zhu, 2022. "A Natural Approximation to the Complete Elliptic Integral of the First Kind," Mathematics, MDPI, vol. 10(9), pages 1-8, April.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:9:p:1472-:d:803518
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    References listed on IDEAS

    as
    1. Zhen-Hang Yang & Jing-Feng Tian & Ya-Ru Zhu, 2020. "A Rational Approximation for the Complete Elliptic Integral of the First Kind," Mathematics, MDPI, vol. 8(4), pages 1-9, April.
    2. Zhi-Jun Guo & Yu-Ming Chu & Ying-Qing Song & Xiao-Jing Tao, 2014. "Sharp Bounds for Neuman Means by Harmonic, Arithmetic, and Contraharmonic Means," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-8, July.
    3. Jonathan M. Borwein & Marc Chamberland, 2007. "Integer Powers of Arcsin," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2007, pages 1-10, May.
    4. Wang, Miao-Kun & Chu, Yu-Ming & Song, Ying-Qing, 2016. "Asymptotical formulas for Gaussian and generalized hypergeometric functions," Applied Mathematics and Computation, Elsevier, vol. 276(C), pages 44-60.
    Full references (including those not matched with items on IDEAS)

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