Abstract
In this paper, we obtained a simple rational approximation for \({\mathcal {K}}_{a}(r)\):
holds for all \(r\in (0,1),\) where \({\mathcal {K}}_{a}(r)=\) \(\frac{\pi }{2}F\left( a,1-a;1;r^{2}\right) =\frac{\pi }{2}\sum _{n=0}^{\infty }\frac{ \left( a\right) _{n}\left( 1-a\right) _{n}}{(n!)^{2}}r^{2n}\) is the generalized elliptic integral of the first kind, and \(r\mathbf {^{\prime }=} \sqrt{1-r^{2}}\). In particular, when \({\small a}\) is taken as 1/2, 1/3, 1/4 and 1/6 respectively, we can obtain the specific lower bound of the corresponding function.
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Acknowledgements
The author is thankful to reviewers for careful corrections to and valuable comments on the original version of this paper. This paper is supported by the Natural Science Foundation of China Grants No. 61772025.
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Zhu, L. A simple rational approximation to the generalized elliptic integral of the first kind. RACSAM 115, 89 (2021). https://doi.org/10.1007/s13398-021-01027-1
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DOI: https://doi.org/10.1007/s13398-021-01027-1