DLMF:19.23.E7 (Q6478)

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DLMF:19.23.E7
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    Statements

    Defining formula
    R G ⁡ ( x , y , z ) = 1 4 ⁢ ∫ 0 ∞ 1 t + x ⁢ t + y ⁢ t + z ⁢ ( x t + x + y t + y + z t + z ) ⁢ t ⁢ d t , Carlson-integral-RG 𝑥 𝑦 𝑧 1 4 superscript subscript 0 1 𝑡 𝑥 𝑡 𝑦 𝑡 𝑧 𝑥 𝑡 𝑥 𝑦 𝑡 𝑦 𝑧 𝑡 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle R_{G}\left(x,y,z\right)=\frac{1}{4}\int_{0}^{% \infty}\frac{1}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}}\*\left(\frac{x}{t+x}+\frac{y}{% t+y}+\frac{z}{t+z}\right)t\mathrm{d}t,}}
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    DLMF id
    DLMF:19.23.E7
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    constraint
    x , y , z ∈ ℂ ∖ ( - ∞ , 0 ] 𝑥 𝑦 𝑧 0 {\displaystyle{\displaystyle x,y,z\in\mathbb{C}\setminus(-\infty,0]}}
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    Symbols used
    symmetric elliptic integral of second kind
    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.m2aadec
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    Q10923
    Defining formula
    ℂ {\displaystyle{\displaystyle\mathbb{C}}}
    xml-id
    introduction.Sx4.p1.t1.r1.m2adec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1afdec
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    Q10784
    Defining formula
    ∈ {\displaystyle{\displaystyle\in}}
    xml-id
    introduction.Sx4.p1.t1.r10.m2adec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3afdec
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    Q11244
    Defining formula
    ( a , b ] 𝑎 𝑏 {\displaystyle{\displaystyle(\NVar{a},\NVar{b}]}}
    xml-id
    introduction.Sx4.p2.t1.r1.m3adec
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    Q10785
    Defining formula
    ∖ {\displaystyle{\displaystyle\setminus}}
    xml-id
    introduction.Sx4.p2.t1.r19.m2adec
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