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A Rational Approximation for the Complete Elliptic Integral of the First Kind

Author

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  • Zhen-Hang Yang

    (Engineering Research Center of Intelligent Computing for Complex Energy Systems of Ministry of Education, North China Electric Power University, Yonghua Street 619, Baoding 071003, China
    Zhejiang Society for Electric Power, Hangzhou 310014, China)

  • Jing-Feng Tian

    (Department of Mathematics and Physics, North China Electric Power University, Yonghua Street 619, Baoding 071003, China)

  • Ya-Ru Zhu

    (Department of Mathematics and Physics, North China Electric Power University, Yonghua Street 619, Baoding 071003, China)

Abstract

Let K ( r ) be the complete elliptic integral of the first kind. We present an accurate rational lower approximation for K ( r ) . More precisely, we establish the inequality 2 π K ( r ) > 5 ( r ′ ) 2 + 126 r ′ + 61 61 ( r ′ ) 2 + 110 r ′ + 21 for r ∈ ( 0 , 1 ) , where r ′ = 1 − r 2 . The lower bound is sharp.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:4:p:635-:d:348267
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    References listed on IDEAS

    as
    1. 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.
    2. Yang, Zhen-Hang & Chu, Yu-Ming & Zhang, Wen, 2019. "High accuracy asymptotic bounds for the complete elliptic integral of the second kind," Applied Mathematics and Computation, Elsevier, vol. 348(C), pages 552-564.
    3. Zhen-Hang Yang & Yu-Ming Chu & Xiao-Jing Tao, 2014. "A Double Inequality for the Trigamma Function and Its Applications," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-9, July.
    4. Belzunce, Felix & Ortega, Eva-Maria & Ruiz, Jose M., 2007. "On non-monotonic ageing properties from the Laplace transform, with actuarial applications," Insurance: Mathematics and Economics, Elsevier, vol. 40(1), pages 1-14, January.
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    Citations

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    Cited by:

    1. Ling Zhu, 2022. "A Natural Approximation to the Complete Elliptic Integral of the First Kind," Mathematics, MDPI, vol. 10(9), pages 1-8, April.
    2. Ling Zhu, 2022. "New Lower Bound for the Generalized Elliptic Integral of the First Kind," Mathematics, MDPI, vol. 10(9), pages 1-13, May.

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