DLMF:19.20.E14 (Q6399)

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DLMF:19.20.E14
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    Statements

    Defining formula
    ( q + z ) ⁢ R J ⁡ ( x , y , z , - q ) = ( p - z ) ⁢ R J ⁡ ( x , y , z , p ) - 3 ⁢ R F ⁡ ( x , y , z ) + 3 ⁢ ( x ⁢ y ⁢ z x ⁢ y + p ⁢ q ) 1 / 2 ⁢ R C ⁡ ( x ⁢ y + p ⁢ q , p ⁢ q ) , 𝑞 𝑧 Carlson-integral-RJ 𝑥 𝑦 𝑧 𝑞 𝑝 𝑧 Carlson-integral-RJ 𝑥 𝑦 𝑧 𝑝 3 Carlson-integral-RF 𝑥 𝑦 𝑧 3 superscript 𝑥 𝑦 𝑧 𝑥 𝑦 𝑝 𝑞 1 2 Carlson-integral-RC 𝑥 𝑦 𝑝 𝑞 𝑝 𝑞 {\displaystyle{\displaystyle(q+z)R_{J}\left(x,y,z,-q\right)=(p-z)R_{J}\left(x,% y,z,p\right)-3R_{F}\left(x,y,z\right)+3\left(\frac{xyz}{xy+pq}\right)^{1/2}R_{% C}\left(xy+pq,pq\right),}}
    0 references
    DLMF id
    DLMF:19.20.E14
    0 references
    Symbols used
    Carlson’s combination of inverse circular and inverse hyperbolic functions
    Defining formula
    R C ⁡ ( x , y ) Carlson-integral-RC 𝑥 𝑦 {\displaystyle{\displaystyle R_{C}\left(\NVar{x},\NVar{y}\right)}}
    xml-id
    C19.S2.E17.m2aedec
    0 references
    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.m2afdec
    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.m2ahdec
    0 references
     
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