DLMF:19.23.E10 (Q6481)

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DLMF:19.23.E10
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    Statements

    Defining formula
    R - a ⁡ ( 𝐛 ; 𝐳 ) = 1 B ⁡ ( a , a ′ ) ⁢ ∫ 0 1 u a - 1 ⁢ ( 1 - u ) a ′ - 1 ⁢ ∏ j = 1 n ( 1 - u + u ⁢ z j ) - b j ⁢ d u , Carlson-integral-R 𝑎 𝐛 𝐳 1 Euler-Beta 𝑎 superscript 𝑎 ′ superscript subscript 0 1 superscript 𝑢 𝑎 1 superscript 1 𝑢 superscript 𝑎 ′ 1 superscript subscript product 𝑗 1 𝑛 superscript 1 𝑢 𝑢 subscript 𝑧 𝑗 subscript 𝑏 𝑗 𝑢 {\displaystyle{\displaystyle R_{-a}\left(\mathbf{b};\mathbf{z}\right)=\frac{1}% {\mathrm{B}\left(a,a^{\prime}\right)}\int_{0}^{1}u^{a-1}(1-u)^{a^{\prime}-1}\*% \prod_{j=1}^{n}(1-u+uz_{j})^{-b_{j}}\mathrm{d}u,}}
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    DLMF id
    DLMF:19.23.E10
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    constraint
    z j ∈ ℂ ∖ ( - ∞ , 0 ] subscript 𝑧 𝑗 0 {\displaystyle{\displaystyle z_{j}\in\mathbb{C}\setminus(-\infty,0]}}
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    a + a ′ = ∑ j = 1 n b j 𝑎 superscript 𝑎 ′ superscript subscript 𝑗 1 𝑛 subscript 𝑏 𝑗 {\displaystyle{\displaystyle a+a^{\prime}=\sum_{j=1}^{n}b_{j}}}
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    z j ∈ ℂ ∖ ( - ∞ , 0 ] subscript 𝑧 𝑗 ℂ 0 {\displaystyle{\displaystyle z_{j}\in\mathbb{C}\setminus(-\infty,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.m2abdec
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    beta function
    Defining formula
    B ⁡ ( a , b ) Euler-Beta 𝑎 𝑏 {\displaystyle{\displaystyle\mathrm{B}\left(\NVar{a},\NVar{b}\right)}}
    xml-id
    C5.S12.E1.m2aadec
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    Q10923
    Defining formula
    ℂ {\displaystyle{\displaystyle\mathbb{C}}}
    xml-id
    introduction.Sx4.p1.t1.r1.m2aadec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aidec
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    Q10784
    Defining formula
    ∈ {\displaystyle{\displaystyle\in}}
    xml-id
    introduction.Sx4.p1.t1.r10.m2aadec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aidec
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