DLMF:5.13.E5 (Q2162)

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DLMF:5.13.E5
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    Statements

    Defining formula
    1 4 ⁢ π ⁢ ∫ - ∞ ∞ ∏ k = 1 4 Γ ⁡ ( a k + i ⁢ t ) ⁢ Γ ⁡ ( a k - i ⁢ t ) Γ ⁡ ( 2 ⁢ i ⁢ t ) ⁢ Γ ⁡ ( - 2 ⁢ i ⁢ t ) ⁢ d t = ∏ 1 ≤ j < k ≤ 4 Γ ⁡ ( a j + a k ) Γ ⁡ ( a 1 + a 2 + a 3 + a 4 ) , 1 4 𝜋 superscript subscript superscript subscript product 𝑘 1 4 Euler-Gamma subscript 𝑎 𝑘 𝑖 𝑡 Euler-Gamma subscript 𝑎 𝑘 𝑖 𝑡 Euler-Gamma 2 𝑖 𝑡 Euler-Gamma 2 𝑖 𝑡 𝑡 subscript product 1 𝑗 𝑘 4 Euler-Gamma subscript 𝑎 𝑗 subscript 𝑎 𝑘 Euler-Gamma subscript 𝑎 1 subscript 𝑎 2 subscript 𝑎 3 subscript 𝑎 4 {\displaystyle{\displaystyle\frac{1}{4\pi}\int_{-\infty}^{\infty}\frac{\prod_{% k=1}^{4}\Gamma\left(a_{k}+it\right)\Gamma\left(a_{k}-it\right)}{\Gamma\left(2% it\right)\Gamma\left(-2it\right)}\mathrm{d}t=\frac{\prod_{1\leq j<k\leq 4}% \Gamma\left(a_{j}+a_{k}\right)}{\Gamma\left(a_{1}+a_{2}+a_{3}+a_{4}\right)},}}
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    DLMF id
    DLMF:5.13.E5
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    constraint
    ℜ ⁡ ( a k ) > 0 subscript 𝑎 𝑘 0 {\displaystyle{\displaystyle\Re(a_{k})>0}}
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    ℜ ⁡ ( a k ) > 0 subscript 𝑎 𝑘 0 {\displaystyle{\displaystyle\Re(a_{k})>0}}
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    k = 1 , 2 , 3 , 4 𝑘 1 2 3 4 {\displaystyle{\displaystyle k=1,2,3,4}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2addec
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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.m2acdec
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