DLMF:35.3.E2 (Q9778): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
Defining formula:

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

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1aadec
Property / Symbols used
 
Property / Symbols used: Q12719 / rank
 
Normal rank
Property / Symbols used: Q12719 / qualifier
 
Defining formula:

𝐗 𝐗 {\displaystyle{\displaystyle\mathbf{X}}}

\mathbf{X}
Property / Symbols used: Q12719 / qualifier
 
xml-id: C35.S1.XMD12.m1adec
Property / Symbols used
 
Property / Symbols used: Q12720 / rank
 
Normal rank
Property / Symbols used: Q12720 / qualifier
 
Defining formula:

𝛀 𝛀 {\displaystyle{\displaystyle{\boldsymbol{\Omega}}}}

{\boldsymbol{\Omega}}
Property / Symbols used: Q12720 / qualifier
 
xml-id: C35.S1.XMD13.m1adec
Property / Symbols used
 
Property / Symbols used: Q12727 / rank
 
Normal rank
Property / Symbols used: Q12727 / qualifier
 
Defining formula:

j 𝑗 {\displaystyle{\displaystyle j}}

j
Property / Symbols used: Q12727 / qualifier
 
xml-id: C35.S1.XMD4.m1dec
Property / Symbols used
 
Property / Symbols used: Q12723 / rank
 
Normal rank
Property / Symbols used: Q12723 / qualifier
 
Defining formula:

m 𝑚 {\displaystyle{\displaystyle m}}

m
Property / Symbols used: Q12723 / qualifier
 
xml-id: C35.S1.XMD8.m1adec

Latest revision as of 14:48, 2 January 2020

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DLMF:35.3.E2
No description defined

    Statements

    Defining formula
    Γ m ⁡ ( s 1 , … , s m ) = ∫ 𝛀 etr ⁡ ( - 𝐗 ) ⁢ | 𝐗 | s m - 1 2 ⁢ ( m + 1 ) ⁢ ∏ j = 1 m - 1 | ( 𝐗 ) j | s j - s j + 1 ⁢ d 𝐗 , multivariate-Euler-Gamma 𝑚 subscript 𝑠 1 … subscript 𝑠 𝑚 subscript 𝛀 exponential-trace 𝐗 superscript 𝐗 subscript 𝑠 𝑚 1 2 𝑚 1 superscript subscript product 𝑗 1 𝑚 1 superscript subscript 𝐗 𝑗 subscript 𝑠 𝑗 subscript 𝑠 𝑗 1 𝐗 {\displaystyle{\displaystyle\Gamma_{m}\left(s_{1},\dots,s_{m}\right)=\int_{% \boldsymbol{\Omega}}\mathrm{etr}\left(-\mathbf{X}\right)|\mathbf{X}|^{s_{m}-% \frac{1}{2}(m+1)}\prod_{j=1}^{m-1}|(\mathbf{X})_{j}|^{s_{j}-s_{j+1}}\mathrm{d}% \mathbf{X},}}
    0 references
    DLMF id
    DLMF:35.3.E2
    0 references
    constraint
    s j ∈ ℂ subscript 𝑠 𝑗 {\displaystyle{\displaystyle s_{j}\in\mathbb{C}}}
    0 references
    ℜ ⁡ ( s j ) > 1 2 ⁢ ( j - 1 ) subscript 𝑠 𝑗 1 2 𝑗 1 {\displaystyle{\displaystyle\Re(s_{j})>\frac{1}{2}(j-1)}}
    0 references
    s j ∈ ℂ subscript 𝑠 𝑗 ℂ {\displaystyle{\displaystyle s_{j}\in\mathbb{C}}}
    0 references
    ℜ ⁡ ( s j ) > 1 2 ⁢ ( j - 1 ) subscript 𝑠 𝑗 1 2 𝑗 1 {\displaystyle{\displaystyle\Re(s_{j})>\frac{1}{2}(j-1)}}
    0 references
    j = 1 , … , m 𝑗 1 … 𝑚 {\displaystyle{\displaystyle j=1,\dots,m}}
    0 references
    Symbols used
    Q10923
    Defining formula
    ℂ {\displaystyle{\displaystyle\mathbb{C}}}
    xml-id
    introduction.Sx4.p1.t1.r1.m2adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aadec
    0 references
    Q10784
    Defining formula
    ∈ {\displaystyle{\displaystyle\in}}
    xml-id
    introduction.Sx4.p1.t1.r10.m2adec
    0 references
    Q12718
    Defining formula
    etr ⁡ ( 𝐗 ) exponential-trace 𝐗 {\displaystyle{\displaystyle\mathrm{etr}\left(\NVar{\mathbf{X}}\right)}}
    xml-id
    C35.S1.p3.t1.r10.m3aadec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aadec
    0 references
    Q12725
    Defining formula
    Γ m ⁡ ( a ) multivariate-Euler-Gamma 𝑚 𝑎 {\displaystyle{\displaystyle\Gamma_{\NVar{m}}\left(\NVar{a}\right)}}
    xml-id
    C35.S3.SS1.m1aadec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aadec
    0 references
    Q12719
    Defining formula
    𝐗 𝐗 {\displaystyle{\displaystyle\mathbf{X}}}
    xml-id
    C35.S1.XMD12.m1adec
    0 references
    Q12720
    Defining formula
    𝛀 𝛀 {\displaystyle{\displaystyle{\boldsymbol{\Omega}}}}
    xml-id
    C35.S1.XMD13.m1adec
    0 references
    Q12727
    Defining formula
    j 𝑗 {\displaystyle{\displaystyle j}}
    xml-id
    C35.S1.XMD4.m1dec
    0 references
    Q12723
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C35.S1.XMD8.m1adec
    0 references
     
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