DLMF:5.20.E5 (Q2210): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11304 / rank
 
Normal rank
Property / Symbols used: Q11304 / qualifier
 
Defining formula:

β = 1 / ( k ⁢ T ) 𝛽 1 Boltzmann-constant 𝑇 {\displaystyle{\displaystyle\beta=1/(kT)}}

\beta=1/(\BoltzmannConstant T)
Property / Symbols used: Q11304 / qualifier
 
xml-id: C5.S20.XMD5.m1adec

Latest revision as of 01:12, 2 January 2020

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DLMF:5.20.E5
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    Statements

    Defining formula
    ψ n ⁢ ( β ) = 1 ( 2 ⁢ π ) n ⁢ ∫ [ - π , π ] n e - β ⁢ W ⁢ d θ 1 ⁢ ⋯ ⁢ d θ n = Γ ⁡ ( 1 + 1 2 ⁢ n ⁢ β ) ⁢ ( Γ ⁡ ( 1 + 1 2 ⁢ β ) ) - n . subscript 𝜓 𝑛 𝛽 1 superscript 2 𝜋 𝑛 subscript superscript 𝜋 𝜋 𝑛 superscript 𝑒 𝛽 𝑊 subscript 𝜃 1 ⋯ subscript 𝜃 𝑛 Euler-Gamma 1 1 2 𝑛 𝛽 superscript Euler-Gamma 1 1 2 𝛽 𝑛 {\displaystyle{\displaystyle\psi_{n}(\beta)=\frac{1}{(2\pi)^{n}}\int_{[-\pi,% \pi]^{n}}e^{-\beta W}\mathrm{d}\theta_{1}\cdots\mathrm{d}\theta_{n}=\Gamma% \left(1+\tfrac{1}{2}n\beta\right)(\Gamma\left(1+\tfrac{1}{2}\beta\right))^{-n}% .}}
    0 references
    DLMF id
    DLMF:5.20.E5
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aadec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
    0 references
    Q10815
    Defining formula
    [ a , b ] 𝑎 𝑏 {\displaystyle{\displaystyle[\NVar{a},\NVar{b}]}}
    xml-id
    introduction.Sx4.p1.t1.r30.m6adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aadec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aadec
    0 references
    Q11274
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C5.S1.XMD3.m1ddec
    0 references
    Q11300
    Defining formula
    W 𝑊 {\displaystyle{\displaystyle W}}
    xml-id
    C5.S20.XMD1.m1ddec
    0 references
    Q11303
    Defining formula
    ψ n ⁢ ( β ) subscript 𝜓 𝑛 𝛽 {\displaystyle{\displaystyle\psi_{n}(\beta)}}
    xml-id
    C5.S20.XMD4.m1adec
    0 references
    Q11304
    Defining formula
    β = 1 / ( k ⁢ T ) 𝛽 1 Boltzmann-constant 𝑇 {\displaystyle{\displaystyle\beta=1/(kT)}}
    xml-id
    C5.S20.XMD5.m1adec
    0 references
     
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