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New Lower Bound for the Generalized Elliptic Integral of the First Kind

Author

Listed:
  • Ling Zhu

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

Abstract

In this paper, we obtain a new simple rational approximation for K a ( r ) : the inequality 2 K a ( r ) / π > g 2 r ′ / g 1 r ′ holds for all r ∈ ( 0 , 1 ) , where K a ( r ) is the generalized elliptic integral of the first kind, r ′ = 1 − r 2 , g 1 r ′ and g 2 r ′ are specific primary and quadratic polynomials about r ′ , respectively. In particular, when a is taken as 1/2, 1/3, 1/4, 1/5 and 1/6 respectively, we can obtain some new specific lower bounds of the corresponding functions.

Suggested Citation

  • Ling Zhu, 2022. "New Lower Bound for the Generalized Elliptic Integral of the First Kind," Mathematics, MDPI, vol. 10(9), pages 1-13, May.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:9:p:1560-:d:809051
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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. 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.
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