DLMF:35.6.E5 (Q9806): Difference between revisions

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‎Changed an Item: Add constraint
 
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Property / constraint
 

ℜ ⁡ ( b ) > 1 2 ⁢ ( m - 1 ) 𝑏 1 2 𝑚 1 {\displaystyle{\displaystyle\Re(b)>\frac{1}{2}(m-1)}}

\Re(b)>\frac{1}{2}(m-1)
Property / constraint: ℜ ⁡ ( b ) > 1 2 ⁢ ( m - 1 ) 𝑏 1 2 𝑚 1 {\displaystyle{\displaystyle\Re(b)>\frac{1}{2}(m-1)}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: generalized hypergeometric function of matrix argument / rank
 
Normal rank
Property / Symbols used: generalized hypergeometric function of matrix argument / qualifier
 
Defining formula:

F q p ⁡ ( a 1 , … , a p ; b 1 , … , b q ; 𝐓 ) Gauss-hypergeometric-of-matrix-pFq 𝑝 𝑞 subscript 𝑎 1 … subscript 𝑎 𝑝 subscript 𝑏 1 … subscript 𝑏 𝑞 𝐓 {\displaystyle{\displaystyle{{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots% ,a_{p}};\NVar{b_{1},\dots,b_{q}};\NVar{\mathbf{T}}\right)}}

\genhyperFmat{\NVar{p}}{\NVar{q}}@{\NVar{a_{1},\dots,a_{p}}}{\NVar{b_{1},\dots,b_{q}}}{\NVar{\mathbf{T}}}
Property / Symbols used: generalized hypergeometric function of matrix argument / qualifier
 
xml-id: C35.S8.E1.m2acdec
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
Defining formula:

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

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1abdec
Property / Symbols used
 
Property / Symbols used: Q12718 / rank
 
Normal rank
Property / Symbols used: Q12718 / qualifier
 
Defining formula:

etr ⁡ ( 𝐗 ) exponential-trace 𝐗 {\displaystyle{\displaystyle\mathrm{etr}\left(\NVar{\mathbf{X}}\right)}}

\exptrace@{\NVar{\mathbf{X}}}
Property / Symbols used: Q12718 / qualifier
 
xml-id: C35.S1.p3.t1.r10.m3abdec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
Defining formula:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3abdec
Property / Symbols used
 
Property / Symbols used: Q12725 / rank
 
Normal rank
Property / Symbols used: Q12725 / qualifier
 
Defining formula:

Γ m ⁡ ( a ) multivariate-Euler-Gamma 𝑚 𝑎 {\displaystyle{\displaystyle\Gamma_{\NVar{m}}\left(\NVar{a}\right)}}

\multivarEulerGamma{\NVar{m}}@{\NVar{a}}
Property / Symbols used: Q12725 / qualifier
 
xml-id: C35.S3.SS1.m1abdec
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.m1acdec
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.m1cdec
Property / Symbols used
 
Property / Symbols used: Q12734 / rank
 
Normal rank
Property / Symbols used: Q12734 / qualifier
 
Defining formula:

𝐒 𝐒 {\displaystyle{\displaystyle\mathbf{S}}}

\mathbf{S}
Property / Symbols used: Q12734 / qualifier
 
xml-id: C35.S1.XMD10.m1dec
Property / Symbols used
 
Property / Symbols used: Q12724 / rank
 
Normal rank
Property / Symbols used: Q12724 / qualifier
 
Defining formula:

𝐓 𝐓 {\displaystyle{\displaystyle\mathbf{T}}}

\mathbf{T}
Property / Symbols used: Q12724 / qualifier
 
xml-id: C35.S1.XMD11.m1ddec
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.m1bdec
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: Q12728 / rank
 
Normal rank
Property / Symbols used: Q12728 / qualifier
 
Defining formula:

b 𝑏 {\displaystyle{\displaystyle b}}

b
Property / Symbols used: Q12728 / qualifier
 
xml-id: C35.S1.XMD2.m1cdec
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.m1cdec

Latest revision as of 14:55, 2 January 2020

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English
DLMF:35.6.E5
No description defined

    Statements

    Defining formula
    ∫ 𝛀 etr ⁡ ( - 𝐓𝐗 ) ⁢ | 𝐗 | b - 1 2 ⁢ ( m + 1 ) ⁢ F 1 1 ⁡ ( a b ; 𝐒𝐗 ) ⁢ d 𝐗 = Γ m ⁡ ( b ) ⁢ | 𝐈 - 𝐒𝐓 - 1 | - a ⁢ | 𝐓 | - b , subscript 𝛀 exponential-trace 𝐓𝐗 superscript 𝐗 𝑏 1 2 𝑚 1 Gauss-hypergeometric-of-matrix-pFq 1 1 𝑎 𝑏 𝐒𝐗 𝐗 multivariate-Euler-Gamma 𝑚 𝑏 superscript 𝐈 superscript 𝐒𝐓 1 𝑎 superscript 𝐓 𝑏 {\displaystyle{\displaystyle\int_{\boldsymbol{\Omega}}\mathrm{etr}\left(-% \mathbf{T}\mathbf{X}\right)|\mathbf{X}|^{b-\frac{1}{2}(m+1)}{{}_{1}F_{1}}\left% ({a\atop b};\mathbf{S}\mathbf{X}\right)\mathrm{d}\mathbf{X}=\Gamma_{m}\left(b% \right)|\mathbf{I}-\mathbf{S}\mathbf{T}^{-1}|^{-a}|\mathbf{T}|^{-b},}}
    0 references
    DLMF id
    DLMF:35.6.E5
    0 references
    constraint
    ℜ ⁡ ( b ) > 1 2 ⁢ ( m - 1 ) 𝑏 1 2 𝑚 1 {\displaystyle{\displaystyle\Re(b)>\frac{1}{2}(m-1)}}
    0 references
    𝐓 > 𝐒 𝐓 𝐒 {\displaystyle{\displaystyle\mathbf{T}>\mathbf{S}}}
    0 references
    ℜ ⁡ ( b ) > 1 2 ⁢ ( m - 1 ) 𝑏 1 2 𝑚 1 {\displaystyle{\displaystyle\Re(b)>\frac{1}{2}(m-1)}}
    0 references
    Symbols used
    generalized hypergeometric function of matrix argument
    Defining formula
    F q p ⁡ ( a 1 , … , a p ; b 1 , … , b q ; 𝐓 ) Gauss-hypergeometric-of-matrix-pFq 𝑝 𝑞 subscript 𝑎 1 … subscript 𝑎 𝑝 subscript 𝑏 1 … subscript 𝑏 𝑞 𝐓 {\displaystyle{\displaystyle{{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots% ,a_{p}};\NVar{b_{1},\dots,b_{q}};\NVar{\mathbf{T}}\right)}}
    xml-id
    C35.S8.E1.m2acdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abdec
    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.m3abdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abdec
    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.m1abdec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1acdec
    0 references
    Q12726
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C35.S1.XMD1.m1cdec
    0 references
    Q12734
    Defining formula
    𝐒 𝐒 {\displaystyle{\displaystyle\mathbf{S}}}
    xml-id
    C35.S1.XMD10.m1dec
    0 references
    Q12724
    Defining formula
    𝐓 𝐓 {\displaystyle{\displaystyle\mathbf{T}}}
    xml-id
    C35.S1.XMD11.m1ddec
    0 references
    Q12719
    Defining formula
    𝐗 𝐗 {\displaystyle{\displaystyle\mathbf{X}}}
    xml-id
    C35.S1.XMD12.m1bdec
    0 references
    Q12720
    Defining formula
    𝛀 𝛀 {\displaystyle{\displaystyle{\boldsymbol{\Omega}}}}
    xml-id
    C35.S1.XMD13.m1adec
    0 references
    Q12728
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C35.S1.XMD2.m1cdec
    0 references
    Q12723
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C35.S1.XMD8.m1cdec
    0 references
     
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