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

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
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

Revision as of 14:55, 2 January 2020

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DLMF:35.6.E5
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    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}}}
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    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
     
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