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A new upper bound for the complete elliptic integral of the first kind

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In this paper, we obtained a concise and high precision approximation for \( \mathcal {K}(r)\):

$$\begin{aligned} \frac{2}{\pi }\mathcal {K}(r)<\frac{120\left( 123r^{\prime }+115\right) }{ 10\,874r^{\prime }-437(r^{\prime })^{2}+18\,123}\left[ \frac{\tanh ^{-1}(r)}{ r}\right] \end{aligned}$$

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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Acknowledgements

The author is thankful to the reviewers for their work in adjusting the structure of this article, which has made the new version of the article particularly concise.

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  1. Department of Mathematics, Zhejiang Gongshang University, Hangzhou, China

    Ling Zhu

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Zhu, L. A new upper bound for the complete elliptic integral of the first kind. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117, 125 (2023). https://doi.org/10.1007/s13398-023-01453-3

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