DLMF:19.16.E9 (Q6324)

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DLMF:19.16.E9
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    Statements

    Defining formula
    R - a ⁡ ( 𝐛 ; 𝐳 ) = 1 B ⁡ ( a , a ′ ) ⁢ ∫ 0 ∞ t a ′ - 1 ⁢ ∏ j = 1 n ( t + z j ) - b j ⁢ d t = 1 B ⁡ ( a , a ′ ) ⁢ ∫ 0 ∞ t a - 1 ⁢ ∏ j = 1 n ( 1 + t ⁢ z j ) - b j ⁢ d t , Carlson-integral-R 𝑎 𝐛 𝐳 1 Euler-Beta 𝑎 superscript 𝑎 ′ superscript subscript 0 superscript 𝑡 superscript 𝑎 ′ 1 subscript superscript product 𝑛 𝑗 1 superscript 𝑡 subscript 𝑧 𝑗 subscript 𝑏 𝑗 𝑡 1 Euler-Beta 𝑎 superscript 𝑎 ′ superscript subscript 0 superscript 𝑡 𝑎 1 subscript superscript product 𝑛 𝑗 1 superscript 1 𝑡 subscript 𝑧 𝑗 subscript 𝑏 𝑗 𝑡 {\displaystyle{\displaystyle R_{-a}\left(\mathbf{b};\mathbf{z}\right)=\frac{1}% {\mathrm{B}\left(a,a^{\prime}\right)}\int_{0}^{\infty}t^{a^{\prime}-1}\prod^{n% }_{j=1}(t+z_{j})^{-b_{j}}\mathrm{d}t=\frac{1}{\mathrm{B}\left(a,a^{\prime}% \right)}\int_{0}^{\infty}t^{a-1}\prod^{n}_{j=1}(1+tz_{j})^{-b_{j}}\mathrm{d}t,}}
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    DLMF id
    DLMF:19.16.E9
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    constraint
    b j ∈ ℝ subscript 𝑏 𝑗 {\displaystyle{\displaystyle b_{j}\in\mathbb{R}}}
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    z j ∈ ℂ ∖ ( - ∞ , 0 ] subscript 𝑧 𝑗 0 {\displaystyle{\displaystyle z_{j}\in\mathbb{C}\setminus(-\infty,0]}}
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    b 1 + ⋯ + b n > a > 0 subscript 𝑏 1 ⋯ subscript 𝑏 𝑛 𝑎 0 {\displaystyle{\displaystyle b_{1}+\cdots+b_{n}>a>0}}
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    b j ∈ ℝ subscript 𝑏 𝑗 ℝ {\displaystyle{\displaystyle b_{j}\in\mathbb{R}}}
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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
    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.m2adec
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    Q10923
    Defining formula
    ℂ {\displaystyle{\displaystyle\mathbb{C}}}
    xml-id
    introduction.Sx4.p1.t1.r1.m2adec
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