DLMF:5.14.E7 (Q2169): Difference between revisions

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Property / constraint
 

b > - 1 / n 𝑏 1 𝑛 {\displaystyle{\displaystyle\Re b>-1/n}}

\realpart@@{b}>-1/n
Property / constraint: b > - 1 / n 𝑏 1 𝑛 {\displaystyle{\displaystyle\Re b>-1/n}} / rank
 
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Revision as of 16:06, 30 December 2019

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DLMF:5.14.E7
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    Statements

    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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    b > - 1 / n 𝑏 1 𝑛 {\displaystyle{\displaystyle\Re b>-1/n}}
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