DLMF:35.8.E13 (Q10527)

From MaRDI portal
Revision as of 09:08, 3 March 2024 by Import240303120314 (talk | contribs) (Added link to MaRDI item.)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)





No description defined
Language Label Description Also known as
English
DLMF:35.8.E13
No description defined

    Statements

    DLMF defining formula
    ∫ 𝟎 < 𝐗 < 𝐈 | 𝐗 | a 1 - 1 2 ⁢ ( m + 1 ) ⁢ | 𝐈 - 𝐗 | b 1 - a 1 - 1 2 ⁢ ( m + 1 ) ⁢ F q p ⁡ ( a 2 , … , a p + 1 b 2 , … , b q + 1 ; 𝐓𝐗 ) ⁢ d 𝐗 = 1 B m ⁡ ( b 1 - a 1 , a 1 ) ⁢ F q + 1 p + 1 ⁡ ( a 1 , … , a p + 1 b 1 , … , b q + 1 ; 𝐓 ) , subscript 0 𝐗 𝐈 superscript 𝐗 subscript 𝑎 1 1 2 𝑚 1 superscript 𝐈 𝐗 subscript 𝑏 1 subscript 𝑎 1 1 2 𝑚 1 Gauss-hypergeometric-of-matrix-pFq 𝑝 𝑞 subscript 𝑎 2 … subscript 𝑎 𝑝 1 subscript 𝑏 2 … subscript 𝑏 𝑞 1 𝐓𝐗 𝐗 1 multivariate-Euler-Beta 𝑚 subscript 𝑏 1 subscript 𝑎 1 subscript 𝑎 1 Gauss-hypergeometric-of-matrix-pFq 𝑝 1 𝑞 1 subscript 𝑎 1 … subscript 𝑎 𝑝 1 subscript 𝑏 1 … subscript 𝑏 𝑞 1 𝐓 {\displaystyle{\displaystyle\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I% }}|\mathbf{X}|^{a_{1}-\frac{1}{2}(m+1)}{|\mathbf{I}-\mathbf{X}|}^{b_{1}-a_{1}-% \frac{1}{2}(m+1)}\*{{}_{p}F_{q}}\left({a_{2},\dots,a_{p+1}\atop b_{2},\dots,b_% {q+1}};\mathbf{T}\mathbf{X}\right)\mathrm{d}\mathbf{X}=\frac{1}{\mathrm{B}_{m}% \left(b_{1}-a_{1},a_{1}\right)}{{}_{p+1}F_{q+1}}\left({a_{1},\dots,a_{p+1}% \atop b_{1},\dots,b_{q+1}};\mathbf{T}\right),}}
    0 references
    constraint
    ℜ ⁡ ( b 1 - a 1 ) , ℜ ⁡ ( a 1 ) > 1 2 ⁢ ( m - 1 ) subscript 𝑏 1 subscript 𝑎 1 subscript 𝑎 1 1 2 𝑚 1 {\displaystyle{\displaystyle\Re(b_{1}-a_{1}),\Re(a_{1})>\frac{1}{2}(m-1)}}
    0 references
    Symbols used
    generalized hypergeometric function of matrix argument
    DLMF 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.m2aldec
    0 references
    differential of x
    DLMF defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aadec
    0 references
    integral
    DLMF defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aadec
    0 references
    multivariate beta function
    DLMF 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
    real part
    DLMF defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1acdec
    0 references
    complex variable
    DLMF defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C35.S1.XMD1.m1jdec
    0 references
    space of real symmetric matrices
    DLMF defining formula
    𝐓 𝐓 {\displaystyle{\displaystyle\mathbf{T}}}
    xml-id
    C35.S1.XMD11.m1hdec
    0 references
    real symmetric matrix
    DLMF defining formula
    𝐗 𝐗 {\displaystyle{\displaystyle\mathbf{X}}}
    xml-id
    C35.S1.XMD12.m1adec
    0 references
    complex variable
    DLMF defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C35.S1.XMD2.m1jdec
    0 references
    nonnegative integer
    DLMF defining formula
    p 𝑝 {\displaystyle{\displaystyle p}}
    xml-id
    C35.S1.XMD6.m1fdec
    0 references
    nonnegative integer
    DLMF defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C35.S1.XMD7.m1fdec
    0 references
    positive integer
    DLMF defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C35.S1.XMD8.m1fdec
    0 references

    Identifiers

     
    Retrieved from "https://mardi.schubotz.org/w/index.php?title=Item:Q10527&oldid=24178524"
    Powered by MediaWiki