DLMF:19.22.E9 (Q6447): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Q11826 / rank
 
Normal rank
Property / Symbols used: Q11826 / qualifier
 
Defining formula:

M ⁡ ( a , g ) arithmetic-geometric-mean 𝑎 𝑔 {\displaystyle{\displaystyle M\left(\NVar{a},\NVar{g}\right)}}

\AGM@{\NVar{a}}{\NVar{g}}
Property / Symbols used: Q11826 / qualifier
 
xml-id: C19.S8.SS1.p1.m7aadec

Revision as of 14:00, 2 January 2020

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DLMF:19.22.E9
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    Statements

    Defining formula
    4 π ⁢ R G ⁡ ( 0 , a 0 2 , g 0 2 ) = 1 M ⁡ ( a 0 , g 0 ) ⁢ ( a 0 2 - ∑ n = 0 ∞ 2 n - 1 ⁢ c n 2 ) = 1 M ⁡ ( a 0 , g 0 ) ⁢ ( a 1 2 - ∑ n = 2 ∞ 2 n - 1 ⁢ c n 2 ) , 4 𝜋 Carlson-integral-RG 0 superscript subscript 𝑎 0 2 superscript subscript 𝑔 0 2 1 arithmetic-geometric-mean subscript 𝑎 0 subscript 𝑔 0 superscript subscript 𝑎 0 2 superscript subscript 𝑛 0 superscript 2 𝑛 1 superscript subscript 𝑐 𝑛 2 1 arithmetic-geometric-mean subscript 𝑎 0 subscript 𝑔 0 superscript subscript 𝑎 1 2 superscript subscript 𝑛 2 superscript 2 𝑛 1 superscript subscript 𝑐 𝑛 2 {\displaystyle{\displaystyle\frac{4}{\pi}R_{G}\left(0,a_{0}^{2},g_{0}^{2}% \right)=\frac{1}{M\left(a_{0},g_{0}\right)}\left(a_{0}^{2}-\sum_{n=0}^{\infty}% 2^{n-1}c_{n}^{2}\right)=\frac{1}{M\left(a_{0},g_{0}\right)}\left(a_{1}^{2}-% \sum_{n=2}^{\infty}2^{n-1}c_{n}^{2}\right),}}
    0 references
    DLMF id
    DLMF:19.22.E9
    0 references
    Symbols used
    symmetric elliptic integral of second kind
    Defining formula
    R G ⁡ ( x , y , z ) Carlson-integral-RG 𝑥 𝑦 𝑧 {\displaystyle{\displaystyle R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)}}
    xml-id
    C19.S16.E3.m2aadec
    0 references
    Q11826
    Defining formula
    M ⁡ ( a , g ) arithmetic-geometric-mean 𝑎 𝑔 {\displaystyle{\displaystyle M\left(\NVar{a},\NVar{g}\right)}}
    xml-id
    C19.S8.SS1.p1.m7aadec
    0 references
     
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