DLMF:35.3.E1 (Q9777): Difference between revisions

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Changed an Item: Add constraint
Changed an Item: Add constraint
 
(4 intermediate revisions by the same user not shown)
Property / Symbols used
 
Property / Symbols used: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
Defining formula:

absent {\displaystyle{\displaystyle\Re}}

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1adec
Property / Symbols used
 
Property / Symbols used: Q12726 / rank
 
Normal rank
Property / Symbols used: Q12726 / qualifier
 
Defining formula:

a 𝑎 {\displaystyle{\displaystyle a}}

a
Property / Symbols used: Q12726 / qualifier
 
xml-id: C35.S1.XMD1.m1dec
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.m1dec
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.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.m1dec

Latest revision as of 14:47, 2 January 2020

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DLMF:35.3.E1
No description defined

    Statements

    Γ m ( a ) = 𝛀 etr ( - 𝐗 ) | 𝐗 | a - 1 2 ( m + 1 ) d 𝐗 , multivariate-Euler-Gamma 𝑚 𝑎 subscript 𝛀 exponential-trace 𝐗 superscript 𝐗 𝑎 1 2 𝑚 1 𝐗 {\displaystyle{\displaystyle\Gamma_{m}\left(a\right)=\int_{\boldsymbol{\Omega}% }\mathrm{etr}\left(-\mathbf{X}\right)|\mathbf{X}|^{a-\frac{1}{2}(m+1)}\mathrm{% d}\mathbf{X},}}
    0 references
    DLMF:35.3.E1
    0 references
    ( a ) > 1 2 ( m - 1 ) 𝑎 1 2 𝑚 1 {\displaystyle{\displaystyle\Re(a)>\frac{1}{2}(m-1)}}
    0 references
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    C1.S4.SS4.m1adec
    0 references
    etr ( 𝐗 ) exponential-trace 𝐗 {\displaystyle{\displaystyle\mathrm{etr}\left(\NVar{\mathbf{X}}\right)}}
    C35.S1.p3.t1.r10.m3adec
    0 references
    {\displaystyle{\displaystyle\int}}
    C1.S4.SS4.m3adec
    0 references
    Γ m ( a ) multivariate-Euler-Gamma 𝑚 𝑎 {\displaystyle{\displaystyle\Gamma_{\NVar{m}}\left(\NVar{a}\right)}}
    C35.S3.SS1.m1adec
    0 references
    absent {\displaystyle{\displaystyle\Re}}
    C1.S9.E2.m1adec
    0 references
    a 𝑎 {\displaystyle{\displaystyle a}}
    C35.S1.XMD1.m1dec
    0 references
    𝐗 𝐗 {\displaystyle{\displaystyle\mathbf{X}}}
    C35.S1.XMD12.m1dec
    0 references
    𝛀 𝛀 {\displaystyle{\displaystyle{\boldsymbol{\Omega}}}}
    C35.S1.XMD13.m1dec
    0 references
    m 𝑚 {\displaystyle{\displaystyle m}}
    C35.S1.XMD8.m1dec
    0 references