DLMF:5.14.E6 (Q2168): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: gamma function / rank
 
Normal rank
Property / Symbols used: gamma function / qualifier
 
Defining formula:

Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}

\EulerGamma@{\NVar{z}}
Property / Symbols used: gamma function / qualifier
 
xml-id: C5.S2.E1.m2addec

Revision as of 01:06, 2 January 2020

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DLMF:5.14.E6
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    Statements

    Defining formula
    1 ( 2 ⁢ π ) n / 2 ⁢ ∫ ( - ∞ , ∞ ) n | Δ ⁢ ( t 1 , … , t n ) | 2 ⁢ c ⁢ ∏ k = 1 n exp ⁡ ( - 1 2 ⁢ t k 2 ) ⁢ d t k = ∏ k = 1 n Γ ⁡ ( 1 + k ⁢ c ) ( Γ ⁡ ( 1 + c ) ) n , 1 superscript 2 𝜋 𝑛 2 subscript superscript 𝑛 superscript Δ subscript 𝑡 1 … subscript 𝑡 𝑛 2 𝑐 superscript subscript product 𝑘 1 𝑛 1 2 superscript subscript 𝑡 𝑘 2 subscript 𝑡 𝑘 superscript subscript product 𝑘 1 𝑛 Euler-Gamma 1 𝑘 𝑐 superscript Euler-Gamma 1 𝑐 𝑛 {\displaystyle{\displaystyle\frac{1}{(2\pi)^{n/2}}\int_{(-\infty,\infty)^{n}}|% \Delta(t_{1},\dots,t_{n})|^{2c}\prod_{k=1}^{n}\exp\left(-\tfrac{1}{2}t_{k}^{2}% \right)\mathrm{d}t_{k}=\frac{\prod_{k=1}^{n}\Gamma\left(1+kc\right)}{(\Gamma% \left(1+c\right))^{n}},}}
    0 references
    DLMF id
    DLMF:5.14.E6
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    constraint
    ℜ ⁡ c > - 1 / n 𝑐 1 𝑛 {\displaystyle{\displaystyle\Re c>-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.m2addec
    0 references
     
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