DLMF:19.21.E10 (Q7124)

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DLMF:19.21.E10
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    Statements

    DLMF defining formula
    2 ⁢ R G ⁡ ( x , y , z ) = z ⁢ R F ⁡ ( x , y , z ) - 1 3 ⁢ ( x - z ) ⁢ ( y - z ) ⁢ R D ⁡ ( x , y , z ) + x ⁢ y / z , 2 Carlson-integral-RG 𝑥 𝑦 𝑧 𝑧 Carlson-integral-RF 𝑥 𝑦 𝑧 1 3 𝑥 𝑧 𝑦 𝑧 Carlson-integral-RD 𝑥 𝑦 𝑧 𝑥 𝑦 𝑧 {\displaystyle{\displaystyle 2R_{G}\left(x,y,z\right)=zR_{F}\left(x,y,z\right)% -\tfrac{1}{3}(x-z)(y-z)R_{D}\left(x,y,z\right)+\sqrt{xy/z},}}
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    constraint
    z ≠ 0 𝑧 0 {\displaystyle{\displaystyle z\neq 0}}
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    Symbols used
    elliptic integral symmetric in only two variables
    DLMF defining formula
    R D ⁡ ( x , y , z ) Carlson-integral-RD 𝑥 𝑦 𝑧 {\displaystyle{\displaystyle R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)}}
    xml-id
    C19.S16.E5.m2agdec
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    symmetric elliptic integral of first kind
    DLMF 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.m2ahdec
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    symmetric elliptic integral of second kind
    DLMF 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.m2abdec
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    Identifiers

     
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