In this paper, we obtain a concise high-precision approximation for $ \mathcal{K}(r) $:
$ \begin{equation*} \frac{2}{\pi }\mathcal{K}(r){\rm{ }}>{\rm{ }}\frac{22\left( r^{\prime }\right) ^{2}+84r^{\prime }+22}{7\left( r^{\prime }\right) ^{3}+57\left( r^{\prime }\right) ^{2}+57r^{\prime }+7}, \end{equation*} $
which 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}} $.
Citation: Ling Zhu. Concise high precision approximation for the complete elliptic integral of the first kind[J]. AIMS Mathematics, 2021, 6(10): 10881-10889. doi: 10.3934/math.2021632
In this paper, we obtain a concise high-precision approximation for $ \mathcal{K}(r) $:
$ \begin{equation*} \frac{2}{\pi }\mathcal{K}(r){\rm{ }}>{\rm{ }}\frac{22\left( r^{\prime }\right) ^{2}+84r^{\prime }+22}{7\left( r^{\prime }\right) ^{3}+57\left( r^{\prime }\right) ^{2}+57r^{\prime }+7}, \end{equation*} $
which 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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Ling Zhu. Concise high precision approximation for the complete elliptic integral of the first kind[J]. AIMS Mathematics, 2021, 6(10): 10881-10889. doi: 10.3934/math.2021632
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