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

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Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
test:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2aadec
Property / Symbols used
 
Property / Symbols used: Q10815 / rank
 
Normal rank
Property / Symbols used: Q10815 / qualifier
 
test:

[ a , b ] 𝑎 𝑏 {\displaystyle{\displaystyle[\NVar{a},\NVar{b}]}}

[\NVar{a},\NVar{b}]
Property / Symbols used: Q10815 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r30.m6aadec
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
test:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1aedec
Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
test:

e {\displaystyle{\displaystyle\mathrm{e}}}

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2aadec
Property / Symbols used
 
Property / Symbols used: imaginary unit / rank
 
Normal rank
Property / Symbols used: imaginary unit / qualifier
 
test:

i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}

\iunit
Property / Symbols used: imaginary unit / qualifier
 
xml-id: C1.S9.E1.m2adec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
test:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3aedec
Property / Symbols used
 
Property / Symbols used: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
test:

ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1aadec
Property / Symbols used
 
Property / Symbols used: Q11292 / rank
 
Normal rank
Property / Symbols used: Q11292 / qualifier
 
test:

j 𝑗 {\displaystyle{\displaystyle j}}

j
Property / Symbols used: Q11292 / qualifier
 
xml-id: C5.S1.XMD1.m1adec
Property / Symbols used
 
Property / Symbols used: Q11274 / rank
 
Normal rank
Property / Symbols used: Q11274 / qualifier
 
test:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11274 / qualifier
 
xml-id: C5.S1.XMD3.m1fdec
Property / Symbols used
 
Property / Symbols used: Q11279 / rank
 
Normal rank
Property / Symbols used: Q11279 / qualifier
 
test:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q11279 / qualifier
 
xml-id: C5.S1.XMD4.m1edec
Property / Symbols used
 
Property / Symbols used: Q11283 / rank
 
Normal rank
Property / Symbols used: Q11283 / qualifier
 
test:

b 𝑏 {\displaystyle{\displaystyle b}}

b
Property / Symbols used: Q11283 / qualifier
 
xml-id: C5.S1.XMD9.m1adec

Latest revision as of 00:07, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:5.14.E7
No description defined

    Statements

    test
    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}},}}
    0 references
    DLMF id
    DLMF:5.14.E7
    0 references
    constraint
    ℜ ⁡ b > - 1 / n 𝑏 1 𝑛 {\displaystyle{\displaystyle\Re b>-1/n}}
    0 references
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aedec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
    0 references
    Q10815
    test
    [ a , b ] 𝑎 𝑏 {\displaystyle{\displaystyle[\NVar{a},\NVar{b}]}}
    xml-id
    introduction.Sx4.p1.t1.r30.m6aadec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aedec
    0 references
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
    0 references
    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aedec
    0 references
    Q10811
    test
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aadec
    0 references
    Q11292
    test
    j 𝑗 {\displaystyle{\displaystyle j}}
    xml-id
    C5.S1.XMD1.m1adec
    0 references
    Q11274
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C5.S1.XMD3.m1fdec
    0 references
    Q11279
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C5.S1.XMD4.m1edec
    0 references
    Q11283
    test
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C5.S1.XMD9.m1adec
    0 references
     
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