DLMF:19.22.E12 (Q6451)

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DLMF:19.22.E12
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    Statements

    Defining formula
    R J ⁡ ( 0 , g 0 2 , a 0 2 , p 0 2 ) = 3 ⁢ π 4 ⁢ M ⁡ ( a 0 , g 0 ) ⁢ p 0 2 ⁢ ∑ n = 0 ∞ Q n , Carlson-integral-RJ 0 superscript subscript 𝑔 0 2 superscript subscript 𝑎 0 2 superscript subscript 𝑝 0 2 3 𝜋 4 arithmetic-geometric-mean subscript 𝑎 0 subscript 𝑔 0 superscript subscript 𝑝 0 2 superscript subscript 𝑛 0 subscript 𝑄 𝑛 {\displaystyle{\displaystyle R_{J}\left(0,g_{0}^{2},a_{0}^{2},p_{0}^{2}\right)% =\frac{3\pi}{4M\left(a_{0},g_{0}\right)p_{0}^{2}}\sum_{n=0}^{\infty}Q_{n},}}
    0 references
    DLMF id
    DLMF:19.22.E12
    0 references
    Symbols used
    symmetric elliptic integral of third kind
    Defining formula
    R J ⁡ ( x , y , z , p ) Carlson-integral-RJ 𝑥 𝑦 𝑧 𝑝 {\displaystyle{\displaystyle R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}% \right)}}
    xml-id
    C19.S16.E2.m2abdec
    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.m7acdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2addec
    0 references
    Q11820
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C19.S1.XMD3.m1cdec
    0 references
    Q11865
    Defining formula
    Q n subscript 𝑄 𝑛 {\displaystyle{\displaystyle Q_{n}}}
    xml-id
    C19.S22.XMD1.m1bdec
    0 references
    Q11824
    Defining formula
    a n subscript 𝑎 𝑛 {\displaystyle{\displaystyle a_{n}}}
    xml-id
    C19.S8.XMD1.m1ddec
    0 references
    Q11825
    Defining formula
    g n subscript 𝑔 𝑛 {\displaystyle{\displaystyle g_{n}}}
    xml-id
    C19.S8.XMD2.m1ddec
    0 references
     
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