DLMF:19.22.E4 (Q6439)

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DLMF:19.22.E4
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    Statements

    Defining formula
    ( p ± 2 - p ∓ 2 ) ⁢ R J ⁡ ( 0 , x 2 , y 2 , p 2 ) = 2 ⁢ ( p ± 2 - a 2 ) ⁢ R J ⁡ ( 0 , x ⁢ y , a 2 , p ± 2 ) - 3 ⁢ R F ⁡ ( 0 , x ⁢ y , a 2 ) + 3 ⁢ π / ( 2 ⁢ p ) , superscript subscript 𝑝 plus-or-minus 2 superscript subscript 𝑝 minus-or-plus 2 Carlson-integral-RJ 0 superscript 𝑥 2 superscript 𝑦 2 superscript 𝑝 2 2 superscript subscript 𝑝 plus-or-minus 2 superscript 𝑎 2 Carlson-integral-RJ 0 𝑥 𝑦 superscript 𝑎 2 superscript subscript 𝑝 plus-or-minus 2 3 Carlson-integral-RF 0 𝑥 𝑦 superscript 𝑎 2 3 𝜋 2 𝑝 {\displaystyle{\displaystyle(p_{\pm}^{2}-p_{\mp}^{2})R_{J}\left(0,x^{2},y^{2},% p^{2}\right)=2(p_{\pm}^{2}-a^{2})R_{J}\left(0,xy,a^{2},p_{\pm}^{2}\right)-3R_{% F}\left(0,xy,a^{2}\right)+3\pi/(2p),}}
    0 references
    DLMF id
    DLMF:19.22.E4
    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.m2acdec
    0 references
    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.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    0 references
     
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