DLMF:5.14.E7 (Q2169)

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DLMF:5.14.E7
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    Statements

    Defining formula
    1 ( 2 ⁢ π ) n ⁢ ∫ [ - π , π ] n ∏ 1 ≤ j < k ≤ n | e i ⁢ θ j - e i ⁢ θ k | 2 ⁢ b ⁢ d θ 1 ⁢ ⋯ ⁢ d θ n = Γ ⁡ ( 1 + b ⁢ n ) ( Γ ⁡ ( 1 + b ) ) n , 1 superscript 2 𝜋 𝑛 subscript superscript 𝜋 𝜋 𝑛 subscript product 1 𝑗 𝑘 𝑛 superscript superscript 𝑒 𝑖 subscript 𝜃 𝑗 superscript 𝑒 𝑖 subscript 𝜃 𝑘 2 𝑏 subscript 𝜃 1 ⋯ subscript 𝜃 𝑛 Euler-Gamma 1 𝑏 𝑛 superscript Euler-Gamma 1 𝑏 𝑛 {\displaystyle{\displaystyle\frac{1}{(2\pi)^{n}}\int_{[-\pi,\pi]^{n}}\prod_{1% \leq j<k\leq n}|e^{i\theta_{j}}-e^{i\theta_{k}}|^{2b}\mathrm{d}\theta_{1}% \cdots\mathrm{d}\theta_{n}=\frac{\Gamma\left(1+bn\right)}{(\Gamma\left(1+b% \right))^{n}},}}
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    DLMF id
    DLMF:5.14.E7
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    constraint
    ℜ ⁡ b > - 1 / n 𝑏 1 𝑛 {\displaystyle{\displaystyle\Re b>-1/n}}
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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.m2aedec
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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.m2aadec
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    Q10815
    Defining formula
    [ a , b ] 𝑎 𝑏 {\displaystyle{\displaystyle[\NVar{a},\NVar{b}]}}
    xml-id
    introduction.Sx4.p1.t1.r30.m6aadec
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