Concise high precision approximation for the complete elliptic integral of the first kind

Ling Zhu; Ling Zhu

doi:10.3934/math.2021632

AIMS Mathematics

2021, Volume 6, Issue 10: 10881-10889. doi: 10.3934/math.2021632

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Research article

Concise high precision approximation for the complete elliptic integral of the first kind

Ling Zhu ^,

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

Received: 13 June 2021 Accepted: 19 July 2021 Published: 28 July 2021
MSC : 33E05, 26E60

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 $r^{\prime } = \sqrt{1-r^{2}}$ .

Keywords:

simple bounds,
rational approximation,
complete elliptic integral of the first kind

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

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Abstract

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

which holds for all $r\in (0, 1)$ , where $\mathcal{K}(r)$ is complete elliptic integral of the first kind and $r^{\prime } = \sqrt{1-r^{2}}$ .

1. Introduction

For $r\in (0, 1)$ , Legendre's complete elliptic integrals of the first kind (see ^[1,2]), denoted $\mathcal{K}(r)$ , is defined by

$\begin{equation*} \mathcal{K}(r) = \int_{0}^{\pi /2}\frac{dt}{\sqrt{1-r^{2}\sin ^{2}(t)}}, \end{equation*}$

which is the particular case of the Gaussian hypergeometric function and can be a special power series (^{[3,4,5,6,7,8,9]}):

$\begin{eqnarray} \mathcal{K}(r) & = &\frac{\pi }{2}F\left( \frac{1}{2}, \frac{1}{2} ;1;r^{2}\right) = \frac{\pi }{2}\sum\limits_{n = 0}^{\infty }\frac{\left( \frac{1}{2} \right) _{n}^{2}}{(n!)^{2}}r^{2n} \\ & = &\frac{\pi }{2}\sum\limits_{n = 0}^{\infty }\left[ \frac{(2n-1)!!}{(2n)!!}\right] ^{2}r^{2n} = :\frac{\pi }{2}\sum\limits_{n = 0}^{\infty }W_{n}^{2}r^{2n}. \end{eqnarray}$

(1.1)

Many researchers have obtained the upper and lower bounds of this special function (^{[10,11,12,13,14,15,16,17,18]}). Let's assume that $r{^{\prime } = } \sqrt{1-r^{2}}$ . In this paper, we use the method similar to Padé approximation to obtain the following rational functions of the argument $r {^{\prime }}$

$\begin{eqnarray} \mathcal{F}_{1, 0}(r{^{\prime }}) & = &\frac{3-r{^{\prime }}}{2}, {\rm{ }} \\ \mathcal{F}_{0, 1}(r{^{\prime }}) & = &\frac{2}{r{^{\prime }}+1}, {\rm{ }} \\ \mathcal{F}_{1, 1}(r{^{\prime }}) & = &\frac{r{^{\prime }}+7}{5r {^{\prime }}+3}, \\ \mathcal{F}_{2, 1}(r{^{\prime }}) & = &\frac{-3\left( r{^{\prime } }\right) ^{2}+22r{^{\prime }}+61}{8\left( 7r{^{\prime }} +3\right) }, {\rm{ }} \\ \mathcal{F}_{1, 2}(r{^{\prime }}) & = &{\rm{ }}\frac{8\left( r^{\prime }+1\right) }{3\left( r^{\prime }\right) ^{2}+10r^{\prime }+3}, \\ \mathcal{F}_{2, 2}(r{^{\prime }}) & = &{\rm{ }}\frac{5\left( r{ ^{\prime }}\right) ^{2}+126r{^{\prime }}+61}{61\left( r{ ^{\prime }}\right) ^{2}+110r{^{\prime }}+21}, \\ \mathcal{F}_{3, 2}(r{^{\prime }}) & = &\frac{5579r{^{\prime }} +495\left( r{^{\prime }}\right) ^{2}-35\left( r{^{\prime }} \right) ^{3}+1769}{2\left( 2050r{^{\prime }}+1567\left( r{ ^{\prime }}\right) ^{2}+287\right) }{\rm{ }}, \\ \mathcal{F}_{2, 3}(r{^{\prime }}) & = &\frac{22\left( r^{\prime }\right) ^{2}+84r^{\prime }+22}{7\left( r^{\prime }\right) ^{3}+57\left( r^{\prime }\right) ^{2}+57r^{\prime }+7} \end{eqnarray}$

(1.2)

to approximate this special function $\mathcal{K}(r)$ , and get the following series of results:

$\begin{eqnarray*} \frac{2}{\pi }\mathcal{K}(r)-\mathcal{F}_{1, 0}(r{^{\prime }}) & = &O(r^{4}), {\rm{ }}\frac{2}{\pi }\mathcal{K}(r)-\mathcal{F}_{0, 1}(r{ ^{\prime }}) = O(r^{4}), \\ \frac{2}{\pi }\mathcal{K}(r)-\mathcal{F}_{1, 1}(r{^{\prime }}) & = &O(r^{6}), \\ \frac{2}{\pi }\mathcal{K}(r)-\mathcal{F}_{2, 1}(r{^{\prime }}) & = &O(r^{8}), {\rm{ }}\frac{2}{\pi }\mathcal{K}(r)-\mathcal{F}_{1, 2}(r{ ^{\prime }}) = O(r^{8}), \\ \frac{2}{\pi }\mathcal{K}(r)-\mathcal{F}_{2, 2}(r{^{\prime }}) & = &O(r^{10}), \\ \frac{2}{\pi }\mathcal{K}(r)-\mathcal{F}_{3, 2}(r{^{\prime }}) & = &O(r^{12}), {\rm{ }}\frac{2}{\pi }\mathcal{K}(r)-\mathcal{F}_{2, 3}(r{ ^{\prime }}) = O(r^{12}). \end{eqnarray*}$

From the last formula above, it is not difficult to find that the bound $\mathcal{F}_{2, 3}(r{^{\prime }})$ is quite sharp. The smaller $r$ is, the smaller the error between $2\mathcal{K}(r)/\pi$ and this bound $\mathcal{F}_{2, 3}(r{^{\prime }})$ is. The following is the error analysis table of $\delta (r) = 2\mathcal{K}(r)/\pi -\mathcal{F}_{2, 3}(r {^{\prime }})$ :

r	0.1	0.2	0.3	0.4	0.5	0.99
$\delta (r)$	$4.\, 680\, \times 10^{-18}$	$2.118\times 10^{-14}$	$3.\, 262\times 10^{-12}$	$1.\, 327\, \times 10^{-10}$	$2.747\times 10^{-9}$	$1.969\times 10^{-2}$

which shows that $\mathcal{F}_{2, 3}(r{^{\prime }})$ has a high precision approximation to $2\mathcal{K}(r)/\pi$ .

Since

$\begin{eqnarray*} \mathcal{F}_{2, 3}(r{^{\prime }})-\mathcal{F}_{3, 2}(r{^{\prime } }) & = & \frac{245}{2}\frac{\left( 1-r^{\prime }\right) ^{6}}{ \left( 7r^{\prime }+1\right) \left( r^{\prime }+7\right) \left( r^{\prime }+1\right) \left( 2050r^{\prime }+1567(r^{\prime })^{2}+287\right) } > 0, \\ \mathcal{F}_{3, 2}(r{^{\prime }})-\mathcal{F}_{2, 2}(r{^{\prime } }) & = &\frac{2135}{2}\frac{\left( 1-r^{\prime }\right) ^{5}}{\left( 110r^{\prime }+61(r^{\prime })^{2}+21\right) \left( 2050r^{\prime }+1567(r^{\prime })^{2}+287\right) } > 0, \\ \mathcal{F}_{2, 2}(r{^{\prime }})-\mathcal{F}_{1, 2}(r{^{\prime } }) & = & 15\frac{\left( 1-r^{\prime }\right) ^{4}}{\left( 3r^{\prime }+1\right) \left( r^{\prime }+3\right) \left( 110r^{\prime }+61(r^{\prime })^{2}+21\right) } > 0, \\ \mathcal{F}_{1, 2}(r{^{\prime }})-\mathcal{F}_{2, 1}(r{^{\prime } }) & = &\frac{9}{8}\frac{\left( 1-r^{\prime }\right) ^{4}}{\left( 7r^{\prime }+3\right) \left( r^{\prime }+3\right) \left( 3r^{\prime }+1\right) } > 0, \\ \mathcal{F}_{2, 1}(r{^{\prime }})-\mathcal{F}_{1, 1}(r{^{\prime } }) & = & \frac{15}{8}\frac{\left( 1-r^{\prime }\right) ^{3}}{\left( 5r^{\prime }+3\right) \left( 7r^{\prime }+3\right) } > 0, \\ \mathcal{F}_{1, 1}(r{^{\prime }})-\mathcal{F}_{0, 1}(r{^{\prime } }) & = & \frac{\left( 1-r^{\prime }\right) ^{2}}{\left( r^{\prime }+1\right) \left( 5r^{\prime }+3\right) } > 0, \\ \mathcal{F}_{0, 1}(r{^{\prime }})-\mathcal{F}_{1, 0}(r{^{\prime } }) & = & \frac{1}{2}\frac{\left( 1-r^{\prime }\right) ^{2}}{ r^{\prime }+1} > 0, \end{eqnarray*}$

we have

$\begin{eqnarray} \mathcal{F}_{2, 3}(r{^{\prime }}) & > &\mathcal{F}_{3, 2}(r{ ^{\prime }}) > \mathcal{F}_{2, 2}(r{^{\prime }}) > \mathcal{F}_{1, 2}(r {^{\prime }}) > \mathcal{F}_{2, 1}(r{^{\prime }}) \\ & > &\mathcal{F}_{1, 1}(r{^{\prime }}) > \mathcal{F}_{0, 1}(r{ ^{\prime }}) > \mathcal{F}_{1, 0}(r{^{\prime }}). \end{eqnarray}$

(1.3)

We first show the following result:

Theorem 1.1. The inequality

$\begin{equation} \frac{2}{\pi }\mathcal{K}(r){\rm{ }} > {\rm{ }} \frac{22\left( r^{\prime }\right) ^{2}+84r^{\prime }+22}{7\left( r^{\prime }\right) ^{3}+57\left( r^{\prime }\right) ^{2}+57r^{\prime }+7} = \mathcal{F}_{2, 3}(r^{\prime }) \end{equation}$

(1.4)

holds for all $r\in (0, 1)$ , where $r^{\prime } = \sqrt{1-r^{2}}$ .

Then by $\left(1.3\right)$ and the above theorem, we have

Theorem 1.2. Let $\mathcal{F}_{i, j}(r^{\prime })$ be defined as $\left(1.2\right).$ Then the following inequality chain

$\begin{eqnarray} \frac{2}{\pi }\mathcal{K}(r){\rm{ }} & > &\mathcal{F}_{2, 3}(r^{\prime }) > \mathcal{F}_{3, 2}(r{^{\prime }}) \\ & > &\mathcal{F}_{2, 2}(r{^{\prime }}) > \mathcal{F}_{1, 2}(r{ ^{\prime }}) > \mathcal{F}_{2, 1}(r{^{\prime }}) \\ & > &\mathcal{F}_{1, 1}(r{^{\prime }}) > \mathcal{F}_{0, 1}(r{ ^{\prime }}) > \mathcal{F}_{1, 0}(r{^{\prime }}) \end{eqnarray}$

(1.5)

holds for all $r\in (0, 1)$ , where $r^{\prime } = \sqrt{1-r^{2}}$ .

2. Lemmas

In order to prove our main results we need following lemmas.

Lemma 2.1.

$\begin{equation} H(n) = :W_{n}- \frac{3}{8}\frac{\left( 2n-1\right) \left( 2n-3\right) \left( 2n-5\right) \left( 111n^{2}+401n-1600\right) }{ n^{2}\left( 49n^{4}+2010n^{3}-13\, 763n^{2}+28\, 212n-17\, 804\right) } > 0 \end{equation}$

(2.1)

holds for all $n\geq 35.$

Proof. It is not difficult to verify that the inequality $\left(2.1\right)$ holds when $n = 35$ . If the inequality $\left(2.1\right)$ holds for $n = m > 35$ , i.e

$\begin{equation*} W_{m} > \frac{3}{8}\frac{\left( 2m-1\right) \left( 2m-3\right) \left( 2m-5\right) \left( 111m^{2}+401m-1600\right) }{m^{2}\left( 49m^{4}+2010m^{3}-13\, 763m^{2}+28\, 212m-17\, 804\right) } . \end{equation*}$

$\begin{eqnarray*} W_{m+1} & = &\frac{(2m+1)!!}{(2m+2)!!} = \frac{2m+1}{2m+2}W_{m} \\ & > &\left( \frac{2m+1}{2m+2}\right) \frac{3}{8}\frac{\left( 2m-1\right) \left( 2m-3\right) \left( 2m-5\right) \left( 111m^{2}+401m-1600\right) }{ m^{2}\left( 49m^{4}+2010m^{3}-13\, 763m^{2}+28\, 212m-17\, 804\right) } . \end{eqnarray*}$

In order to complete the proof of $(2.1)$ it suffices to show that

$\begin{eqnarray*} &&\left( \frac{2m+1}{2m+2}\right) \frac{3}{8}\frac{\left( 2m-1\right) \left( 2m-3\right) \left( 2m-5\right) \left( 111m^{2}+401m-1600\right) }{ m^{2}\left( 49m^{4}+2010m^{3}-13\, 763m^{2}+28\, 212m-17\, 804\right) } \\ & > &\frac{3}{8}\frac{\left( 2\left( m+1\right) -1\right) \left( 2\left( m+1\right) -3\right) \left( 2\left( m+1\right) -5\right) \left( 111\left( m+1\right) ^{2}+401\left( m+1\right) -1600\right) }{\left( m+1\right) ^{2} \left[ 49\left( m+1\right) ^{4}+2010\left( m+1\right) ^{3}-13\, 763\left( m+1\right) ^{2}+28\, 212\left( m+1\right) -17\, 804\right] } \\ & = &\frac{3}{8}\frac{\left( 2m-1\right) \left( 2m+1\right) \left( 2m-3\right) \left( 623m+111m^{2}-1088\right) }{\left( m+1\right) ^{2}\left( 6912m-7439m^{2}+2206m^{3}+49m^{4}-1296\right) } , \end{eqnarray*}$

which is

$\begin{eqnarray*} \frac{A}{B} & = :&\frac{\left( 2m-5\right) \left( 111m^{2}+401m-1600\right) }{\left( 2m+2\right) m^{2}\left( 49m^{4}+2010m^{3}-13\, 763m^{2}+28\, 212m-17\, 804\right) } \\ & > &\frac{\left( 623m+111m^{2}-1088\right) }{\left( m+1\right) ^{2}\left( 6912m-7439m^{2}+2206m^{3}+49m^{4}-1296\right) } = :\frac{C}{D}. \end{eqnarray*}$

In fact,

$\begin{eqnarray*} AD-BC & = &5439m^{7}- 202\, 236m^{6}+2504\, 574m^{5}-14\, 753\, 604m^{4}+ 45\, 551\, 571m^{3} \\ &&-72\, 508\, 896m^{2}+51\, 673\, 680m-10\, 368\, 000 \\ & = & 9308\, 275\, 200+137\, 097\, 875\, 040\left( m-20\right) + 47\, 752\, 255\, 764\left( m-20\right) ^{2} \\ &&+6984\, 199\, 251\left( m-20\right) ^{3}+ 545\, 207\, 796\left( m-20\right) ^{4}+23\, 923\, 854\left( m-20\right) ^{5} \\ &&+ 559\, 224\left( m-20\right) ^{6}+5439\left( m-20\right) ^{7} \\ & > &0. \end{eqnarray*}$

Lemma 2.1.(^[17,18]) Let $\{a_{k}\}_{k = 0}^{\infty }$ be a nonnegative real sequence with $a_{m} > 0$ and $\sum_{k = m+1}^{\infty }a_{k} > 0$ , and

$\begin{equation*} S(t) = -\sum\limits_{k = 0}^{m}a_{k}t^{k}+\sum\limits_{k = m+1}^{\infty }a_{k}t^{k} \end{equation*}$

be a convergent power series on the interval $(0, r)$ $(r > 0)$ . Then the following statements are true:

$(1)$ If $S(r^{-})\leq 0$ , then $S(t) < 0$ for all $t\in (0, r)$ ;

$(2)$ If $S(r^{-}) > 0$ , then there exists $t_{0}\in (0, r)$ such that $S(t) < 0$ for $t\in (0, t_{0})$ and $S(t) > 0$ for $t\in (t_{0}, r)$ .

3. Proof of Theorem 1.1

Let

$\begin{eqnarray*} f(r) & = &\left[ 49\left( r{^{\prime }}\right) ^{6}-2451\left( r{ ^{\prime }}\right) ^{4}+2451\left( r{^{\prime }}\right) ^{2}-49\right] \frac{2}{\pi }\mathcal{K}(r) \\ &&{-}\left[ 154\left( r{^{\prime }}\right) ^{5}-666\left( r {^{\prime }}\right) ^{4}-3380\left( r{^{\prime }}\right) ^{3}+3380\left( r{^{\prime }}\right) ^{2}+666\left( r{^{\prime }}\right) - 154\right] . \end{eqnarray*}$

Then

$\begin{eqnarray*} f(r) & = &\left[ 49\left( r{^{\prime }}\right) ^{6}-2451\left( r{ ^{\prime }}\right) ^{4}+2451\left( r{^{\prime }}\right) ^{2}-49\right] \frac{2}{\pi }\mathcal{K}(r) \\ &&{-} \left[ 666\sqrt{1-r^{2}}-3380\left( 1-r^{2}\right) ^{ \frac{3}{2}}+154\left( 1-r^{2}\right) ^{\frac{5}{2}}- 2048r^{2}-666r^{4}+2560\right] \\ & = & \left( -49r^{6}-2304r^{4}+2304r^{2}\right) \sum\limits_{n = 0}^{\infty }W_{n}^{2}r^{2n}-666\sum\limits_{n = 7}^{\infty }\frac{\left( -\frac{1}{2} \right) _{n}}{n!}r^{2n}+3380\sum\limits_{n = 7}^{\infty }\frac{\left( -\frac{3 }{2}\right) _{n}}{n!}r^{2n} \\ &&-154\sum\limits_{n = 7}^{\infty }\frac{\left( -\frac{5}{2}\right) _{n}}{n!} r^{2n}+ 2048r^{2}+666r^{4}-2560 \\ & = & -49\sum\limits_{n = 4}^{\infty }W_{n}^{2}r^{2n+6}-2304\sum\limits_{n = 5}^{\infty }W_{n}^{2}r^{2n+4}+2304\sum\limits_{n = 6}^{\infty }W_{n}^{2}r^{2n+2} \\ &&-666\sum\limits_{n = 7}^{\infty }\frac{\left( -\frac{1}{2}\right) _{n}}{n!} r^{2n} +3380\sum\limits_{n = 7}^{\infty }\frac{\left( -\frac{3}{2} \right) _{n}}{n!}r^{2n}-154\sum\limits_{n = 7}^{\infty }\frac{\left( -\frac{5 }{2}\right) _{n}}{n!}r^{2n} \end{eqnarray*}$

$\begin{eqnarray*} & = & -49\sum\limits_{n = 7}^{\infty }W_{n-3}^{2}r^{2n}-2304\sum\limits_{n = 7}^{\infty }W_{n-2}^{2}r^{2n}+2304\sum\limits_{n = 7}^{\infty }W_{n-1}^{2}r^{2n} \\ &&-666\sum\limits_{n = 7}^{\infty }\frac{\left( -\frac{1}{2}\right) _{n}}{n!} r^{2n}+3380\sum\limits_{n = 7}^{\infty }\frac{\left( -\frac{3}{2}\right) _{n} }{n!}r^{2n}-154\sum\limits_{n = 7}^{\infty }\frac{\left( -\frac{5}{2}\right) _{n}}{n!}r^{2n} \\ & = & -49\sum\limits_{n = 7}^{\infty }W_{n-3}^{2}r^{2n}-2304\sum\limits_{n = 7}^{\infty }W_{n-2}^{2}r^{2n}+2304\sum\limits_{n = 7}^{\infty }W_{n-1}^{2}r^{2n} \\ &&+3380\left( -\frac{3}{2}\right) \left( -\frac{1}{2}\right) \sum\limits_{n = 7}^{\infty }\frac{\left( \frac{1}{2}\right) _{n-2}}{n!} r^{2n}-666\left( -\frac{1}{2}\right) \sum\limits_{n = 7}^{\infty }\frac{ \left( \frac{1}{2}\right) _{n-1}}{n!}r^{2n} \\ &&-154\left( -\frac{5}{2}\right) \left( -\frac{3}{2}\right) \left( -\frac{1}{ 2}\right) \sum\limits_{n = 7}^{\infty }\frac{\left( \frac{1}{2}\right) _{n-3} }{n!}r^{2n} \\ & = & -49\sum\limits_{n = 7}^{\infty }W_{n-3}^{2}r^{2n}-2304\sum\limits_{n = 7}^{\infty }W_{n-2}^{2}r^{2n}+2304\sum\limits_{n = 7}^{\infty }W_{n-1}^{2}r^{2n}+ 2535\sum\limits_{n = 7}^{\infty }\frac{\left( \frac{1}{2}\right) _{n-2}}{n!} r^{2n} \\ &&+ 333\sum\limits_{n = 7}^{\infty }\frac{\left( \frac{1}{2} \right) _{n-1}}{n!}r^{2n}+ \frac{1155}{4}\sum\limits_{n = 7}^{ \infty }\frac{\left( \frac{1}{2}\right) _{n-3}}{n!}r^{2n} \\ & = :&\sum\limits_{n = 7}^{\infty }c_{n}r^{2n}, \end{eqnarray*}$

where

$\begin{equation*} c_{n} = -49W_{n-3}^{2}-2304W_{n-2}^{2}+2304W_{n-1}^{2}+ 2535\frac{\left( \frac{1}{2}\right) _{n-2}}{n!}+ 333\frac{\left( \frac{1}{2}\right) _{n-1}}{n!}+ \frac{1155}{4}\frac{\left( \frac{1 }{2}\right) _{n-3}}{n!}. \end{equation*}$

$\begin{equation*} W_{n-1} = \frac{2n}{2n-1}W_{n}\rm{ and }\left( \frac{1}{2}\right) _{n} = \left( \frac{1}{2}\right) _{n-1}\left( \frac{1}{2}+\left( n-1\right) \right) \end{equation*}$

we have

$\begin{eqnarray*} c_{n} & = &-49\left( \frac{2n-4}{2n-5}\frac{2n-2}{2n-3}\frac{2n}{2n-1} W_{n}\right) ^{2}-2304\left( \frac{2n-2}{2n-3}\frac{2n}{2n-1}W_{n}\right) ^{2}+2304\left( \frac{2n}{2n-1}W_{n}\right) ^{2} \\ &&+ \frac{333}{\left( \frac{1}{2}+\left( n-1\right) \right) } \frac{\left( \frac{1}{2}\right) _{n}}{n!}+ \frac{2535}{\left( \frac{1}{2}+\left( n-1\right) \right) \left( \frac{1}{2}+\left( n-2\right) \right) } \\ &&+ \frac{1155}{4\left( \frac{1}{2}+\left( n-1\right) \right) \left( \frac{1}{2}+\left( n-2\right) \right) \left( \frac{1}{2}+\left( n-3\right) \right) }\frac{\left( \frac{1}{2}\right) _{n}}{n!} \\ & = & -64n^{2}\frac{28\, 212n-13\, 763n^{2}+2010n^{3}+49n^{4}-17\, 804 }{\left( 2n-3\right) ^{2}\left( 2n-5\right) ^{2}\left( 2n-1\right) ^{2}} W_{n}^{2} +24\frac{401n+111n^{2}-1600}{\left( 2n-3\right) \left( 2n-5\right) \left( 2n-1\right) }W_{n} \\ & = &-64n^{2}\frac{28\, 212n-13\, 763n^{2}+2010n^{3}+49n^{4}-17\, 804}{\left( 2n-3\right) ^{2}\left( 2n-5\right) ^{2}\left( 2n-1\right) ^{2}}W_{n}H(n). \end{eqnarray*}$

It's not difficult to verify that $c_{n} > 0$ for $n = 7, 8, \cdots, 34,$ and according to Lemma 2.1, $c_{n} < 0$ for all $n\geq 35.$

So those coefficients in power series of $-f(r)$ satisfy the conditions of Lemma 2.2, and $-f(1^{-}) = \infty$ . From Lemma 2.2, it follows that there is a unique $r^{\ast }\in \left(0, 1\right)$ such that $-f(r) < 0$ for $r\in (0, r^{\ast })$ and $-f(r) > 0$ for $r\in (r^{\ast }, 1)$ , that is, $\ f(r) > 0$ for $r\in (0, r^{\ast })$ and $f(r) < 0$ for $r\in (r^{\ast }, 1),$ and $r^{\ast }$ is the unique zero of $f(r)$ on $\left(0, 1\right)$ . At the same time, since

$\begin{eqnarray*} f(r) & = &\left[ 7\left( r{^{\prime }}\right) ^{3}-57\left( r{ ^{\prime }}\right) ^{2}+57\left( r{^{\prime }}\right) -7\right] \times \\ &&\left\{ \left[ 7\left( r{^{\prime }}\right) ^{3}+57\left( r{ ^{\prime }}\right) ^{2}+57\left( r{^{\prime }}\right) +7\right] \frac{ 2}{\pi }\mathcal{K}(r)-\left[ 22\left( r{^{\prime }}\right) ^{2}+84\left( r{^{\prime }}\right) +22\right] \right\} , \end{eqnarray*}$

and the function

$\begin{eqnarray*} g(r) & = :&7\left( r{^{\prime }}\right) ^{3}-57\left( r{^{\prime }}\right) ^{2}+57\left( r{^{\prime }}\right) -7 \\ & = &\frac{ 49\left( r{^{\prime }}\right) ^{6}-2451\left( r {^{\prime }}\right) ^{4}+2451\left( r{^{\prime }}\right) ^{2}-49}{7\left( r{^{\prime }}\right) ^{3}+57\left( r{^{\prime }}\right) ^{2}+57\left( r{^{\prime }}\right) +7} \\ & = &\frac{-49r^{6}-2304r^{4}+2304r^{2}}{7\left( r{^{\prime }}\right) ^{3}+57\left( r{^{\prime }}\right) ^{2}+57\left( r{^{\prime }} \right) +7} \\ & = &\frac{-49r^{2}\left( r-4\sqrt{3}/7\right) \left( r+4\sqrt{3}/7\right) \left( r^{2}+48\right) }{7\left( r{^{\prime }}\right) ^{3}+57\left( r{^{\prime }}\right) ^{2}+57\left( r{^{\prime }} \right) +7} \end{eqnarray*}$

has a unique zero $4\sqrt{3}/7$ on $\left(0, 1\right)$ , which leads to $r^{\ast } = 4\sqrt{3}/7$ . Clearly, $g(r) > 0$ for $r\in (0, r^{\ast })$ and $g(r) < 0$ for $r\in (r^{\ast }, 1)$ . That is to say, the two functions $f(r)$ and $g(r)$ have the same sign on both sides of the point $r^{\ast }$ , so we have

$\begin{eqnarray*} &&\frac{2}{\pi }\mathcal{K}(r)-\frac{22\left( r{^{\prime }}\right) ^{2}+84\left( r{^{\prime }}\right) +22}{7\left( r{^{\prime }} \right) ^{3}+57\left( r{^{\prime }}\right) ^{2}+57\left( r{ ^{\prime }}\right) +7} \\ & = &\frac{f(r)}{g(r)\left[ 7\left( r{^{\prime }}\right) ^{3}+57\left( r {^{\prime }}\right) ^{2}+57\left( r{^{\prime }}\right) +7 \right] } > 0 \end{eqnarray*}$

holds for all $r\in \left(0, 1\right)$ with $r\neq r^{\ast }$ . But the continuity at $r = r^{\ast }$ of the functions $\mathcal{K}(r)$ and $\mathcal{F }_{2, 3}(r)$ ensures that the inequality (1.4) also holds for $r = r^{\ast }$ .

The proof of Theorem 1.1 is complete.

4. Remark

Recently, Z. H. Yang, J. F. Tian and Y. R. Zhu ^[18] presented a new sharp lower bound for the complete elliptic integral of the first kind as follows:

$\begin{equation} \frac{2}{\pi }\mathcal{K}(r) > \frac{5\left( r{^{\prime }}\right) ^{2}+126r{^{\prime }}+61}{61\left( r{^{\prime }}\right) ^{2}+110r{^{\prime }}+21} = \mathcal{F}_{2, 2}(r{^{\prime }}), \end{equation}$

(4.1)

where $r\in (0, 1)$ and $r^{\prime } = \sqrt{1-r^{2}}$ . It is not difficult to find that the above conclusion is a direct corollary of Theorem 1.2.

5. Conclusions

This paper obtains a new lower bound for $2\mathcal{K}(r)/\pi$ :

$\begin{equation*} \frac{2}{\pi }\mathcal{K}(r){\rm{ }} > {\rm{ }}\frac{22\left( r^{\prime }\right) ^{2}+84r^{\prime }+22}{7\left( r^{\prime }\right) ^{3}+57\left( r^{\prime }\right) ^{2}+57r^{\prime }+7}. \end{equation*}$

This lower bound approximates $2\mathcal{K}(r)/\pi$ with high accuracy.

Acknowledgments

The author is thankful to reviewers for their valuable comments on the original version of this paper.

The research was supported by National Natural Science Foundation of China (Grants No. 61772025).

Conflict of interest

The author declares that he has no conflict of interest.

References

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AIMS Mathematics

Concise high precision approximation for the complete elliptic integral of the first kind

Related Papers:

Abstract

1. Introduction

2. Lemmas

3. Proof of Theorem 1.1

4. Remark

5. Conclusions

Acknowledgments

Conflict of interest

References

This article has been cited by:

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通讯作者: 陈斌, bchen63@163.com

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Catalog

Abstract

1. Introduction

2. Lemmas

3. Proof of Theorem 1.1

4. Remark

5. Conclusions

Acknowledgments

Conflict of interest

References

AIMS Mathematics

Concise high precision approximation for the complete elliptic integral of the first kind

Related Papers:

Abstract

1. Introduction

2. Lemmas

3. Proof of Theorem 1.1

4. Remark

5. Conclusions

Acknowledgments

Conflict of interest

References

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Abstract

1. Introduction

2. Lemmas

3. Proof of Theorem 1.1

4. Remark

5. Conclusions

Acknowledgments

Conflict of interest

References