DLMF:19.22.E8 (Q6446)

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DLMF:19.22.E8
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    Statements

    Defining formula
    2 π ⁢ R F ⁡ ( 0 , a 0 2 , g 0 2 ) = 1 M ⁡ ( a 0 , g 0 ) , 2 𝜋 Carlson-integral-RF 0 superscript subscript 𝑎 0 2 superscript subscript 𝑔 0 2 1 arithmetic-geometric-mean subscript 𝑎 0 subscript 𝑔 0 {\displaystyle{\displaystyle\frac{2}{\pi}R_{F}\left(0,a_{0}^{2},g_{0}^{2}% \right)=\frac{1}{M\left(a_{0},g_{0}\right)},}}
    0 references
    DLMF id
    DLMF:19.22.E8
    0 references
    Symbols used
    symmetric elliptic integral of first kind
    Defining formula
    R F ⁡ ( x , y , z ) Carlson-integral-RF 𝑥 𝑦 𝑧 {\displaystyle{\displaystyle R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)}}
    xml-id
    C19.S16.E1.m2aedec
    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.m7adec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
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    Q11824
    Defining formula
    a n subscript 𝑎 𝑛 {\displaystyle{\displaystyle a_{n}}}
    xml-id
    C19.S8.XMD1.m1dec
    0 references
    Q11825
    Defining formula
    g n subscript 𝑔 𝑛 {\displaystyle{\displaystyle g_{n}}}
    xml-id
    C19.S8.XMD2.m1dec
    0 references
     
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