IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v10y2022i9p1560-d809051.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/10/9/1560/pdf
    Download Restriction: no

    File URL: https://www.mdpi.com/2227-7390/10/9/1560/
    Download Restriction: no
    ---><---

    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    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. 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. Mansour Mahmoud & Hanan Almuashi, 2024. "Two Approximation Formulas for Gamma Function with Monotonic Remainders," Mathematics, MDPI, vol. 12(5), pages 1-15, February.
    4. Tie-Hong Zhao & Wei-Mao Qian & Yu-Ming Chu, 2022. "On approximating the arc lemniscate functions," Indian Journal of Pure and Applied Mathematics, Springer, vol. 53(2), pages 316-329, June.
    5. Zhen-Hang Yang & Jing-Feng Tian & Ya-Ru Zhu, 2020. "New Sharp Bounds for the Modified Bessel Function of the First Kind and Toader-Qi Mean," Mathematics, MDPI, vol. 8(6), pages 1-13, June.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:10:y:2022:i:9:p:1560-:d:809051. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.