DLMF:19.23.E9 (Q6480)

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DLMF:19.23.E9
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    Statements

    Defining formula
    R - a ⁡ ( 𝐛 ; 𝐳 ) = 4 ⁢ Γ ⁡ ( b 1 + b 2 + b 3 ) Γ ⁡ ( b 1 ) ⁢ Γ ⁡ ( b 2 ) ⁢ Γ ⁡ ( b 3 ) ⁢ ∫ 0 π / 2 ∫ 0 π / 2 ( ∑ j = 1 3 z j ⁢ l j 2 ) - a ⁢ ∏ j = 1 3 l j 2 ⁢ b j - 1 ⁢ sin ⁡ θ ⁢ d θ ⁢ d ϕ , Carlson-integral-R 𝑎 𝐛 𝐳 4 Euler-Gamma subscript 𝑏 1 subscript 𝑏 2 subscript 𝑏 3 Euler-Gamma subscript 𝑏 1 Euler-Gamma subscript 𝑏 2 Euler-Gamma subscript 𝑏 3 superscript subscript 0 𝜋 2 superscript subscript 0 𝜋 2 superscript superscript subscript 𝑗 1 3 subscript 𝑧 𝑗 superscript subscript 𝑙 𝑗 2 𝑎 superscript subscript product 𝑗 1 3 superscript subscript 𝑙 𝑗 2 subscript 𝑏 𝑗 1 𝜃 𝜃 italic-ϕ {\displaystyle{\displaystyle R_{-a}\left(\mathbf{b};\mathbf{z}\right)=\frac{4% \Gamma\left(b_{1}+b_{2}+b_{3}\right)}{\Gamma\left(b_{1}\right)\Gamma\left(b_{2% }\right)\Gamma\left(b_{3}\right)}\int_{0}^{\pi/2}\!\!\!\!\int_{0}^{\pi/2}\left% (\sum_{j=1}^{3}z_{j}l_{j}^{2}\right)^{-a}\*\prod_{j=1}^{3}l_{j}^{2b_{j}-1}\sin% \theta\mathrm{d}\theta\mathrm{d}\phi,}}
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    DLMF id
    DLMF:19.23.E9
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    constraint
    ℜ ⁡ z j > 0 subscript 𝑧 𝑗 0 {\displaystyle{\displaystyle\Re z_{j}>0}}
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    b j > 0 subscript 𝑏 𝑗 0 {\displaystyle{\displaystyle b_{j}>0}}
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    ℜ ⁡ z j > 0 subscript 𝑧 𝑗 0 {\displaystyle{\displaystyle\Re z_{j}>0}}
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    Symbols used
    multivariate hypergeometric function
    Defining formula
    R - a ⁡ ( b 1 , … , b n ; z 1 , … , z n ) Carlson-integral-R 𝑎 subscript 𝑏 1 … subscript 𝑏 𝑛 subscript 𝑧 1 … subscript 𝑧 𝑛 {\displaystyle{\displaystyle R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}% };\NVar{z_{1}},\dots,\NVar{z_{n}}\right)}}
    xml-id
    C19.S16.E9.m2aadec
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    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2agdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1ahdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3ahdec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1abdec
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    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2afdec
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    Q11855
    Defining formula
    l 𝑙 {\displaystyle{\displaystyle l}}
    xml-id
    C19.S1.XMD1.m1dec
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