DLMF:35.6.E4 (Q9805)

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DLMF:35.6.E4
No description defined

    Statements

    Defining formula
    F 1 1 ⁡ ( a b ; 𝐓 ) = 1 B m ⁡ ( a , b - a ) ⁢ ∫ 𝟎 < 𝐗 < 𝐈 etr ⁡ ( 𝐓𝐗 ) ⁢ | 𝐗 | a - 1 2 ⁢ ( m + 1 ) ⁢ | 𝐈 - 𝐗 | b - a - 1 2 ⁢ ( m + 1 ) ⁢ d 𝐗 , Gauss-hypergeometric-of-matrix-pFq 1 1 𝑎 𝑏 𝐓 1 multivariate-Euler-Beta 𝑚 𝑎 𝑏 𝑎 subscript 0 𝐗 𝐈 exponential-trace 𝐓𝐗 superscript 𝐗 𝑎 1 2 𝑚 1 superscript 𝐈 𝐗 𝑏 𝑎 1 2 𝑚 1 𝐗 {\displaystyle{\displaystyle{{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)=% \frac{1}{\mathrm{B}_{m}\left(a,b-a\right)}\int\limits_{\boldsymbol{{0}}<% \mathbf{X}<\mathbf{I}}\mathrm{etr}\left(\mathbf{T}\mathbf{X}\right)|\mathbf{X}% |^{a-\frac{1}{2}(m+1)}|\mathbf{I}-\mathbf{X}|^{b-a-\frac{1}{2}(m+1)}\mathrm{d}% \mathbf{X},}}
    0 references
    DLMF id
    DLMF:35.6.E4
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    constraint
    ℜ ⁡ ( a ) , ℜ ⁡ ( b - a ) > 1 2 ⁢ ( m - 1 ) 𝑎 𝑏 𝑎 1 2 𝑚 1 {\displaystyle{\displaystyle\Re(a),\Re(b-a)>\frac{1}{2}(m-1)}}
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    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.m2abdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aadec
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    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
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aadec
    0 references
    multivariate beta function
    Defining formula
    B m ⁡ ( a , b ) multivariate-Euler-Beta 𝑚 𝑎 𝑏 {\displaystyle{\displaystyle\mathrm{B}_{\NVar{m}}\left(\NVar{a},\NVar{b}\right% )}}
    xml-id
    C35.S3.E3.m2adec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1abdec
    0 references
    Q12726
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C35.S1.XMD1.m1bdec
    0 references
    Q12724
    Defining formula
    𝐓 𝐓 {\displaystyle{\displaystyle\mathbf{T}}}
    xml-id
    C35.S1.XMD11.m1cdec
    0 references
    Q12719
    Defining formula
    𝐗 𝐗 {\displaystyle{\displaystyle\mathbf{X}}}
    xml-id
    C35.S1.XMD12.m1adec
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    Q12728
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C35.S1.XMD2.m1bdec
    0 references
    Q12723
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C35.S1.XMD8.m1bdec
    0 references
     
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