DLMF:35.6.E8 (Q9809): Difference between revisions

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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.m1fdec
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.m1gdec
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.m1bdec
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.m1fdec
Property / Symbols used
 
Property / Symbols used: Q12736 / rank
 
Normal rank
Property / Symbols used: Q12736 / qualifier
 
Defining formula:

c 𝑐 {\displaystyle{\displaystyle c}}

c
Property / Symbols used: Q12736 / qualifier
 
xml-id: C35.S1.XMD3.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.m1edec

Latest revision as of 14:56, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:35.6.E8
No description defined

    Statements

    Defining formula
    ∫ 𝛀 | 𝐓 | c - 1 2 ⁢ ( m + 1 ) ⁢ Ψ ⁡ ( a ; b ; 𝐓 ) ⁢ d 𝐓 = Γ m ⁡ ( c ) ⁢ Γ m ⁡ ( a - c ) ⁢ Γ m ⁡ ( c - b + 1 2 ⁢ ( m + 1 ) ) Γ m ⁡ ( a ) ⁢ Γ m ⁡ ( a - b + 1 2 ⁢ ( m + 1 ) ) , subscript 𝛀 superscript 𝐓 𝑐 1 2 𝑚 1 confluent-hypergeometric-of-matrix-Psi 𝑎 𝑏 𝐓 𝐓 multivariate-Euler-Gamma 𝑚 𝑐 multivariate-Euler-Gamma 𝑚 𝑎 𝑐 multivariate-Euler-Gamma 𝑚 𝑐 𝑏 1 2 𝑚 1 multivariate-Euler-Gamma 𝑚 𝑎 multivariate-Euler-Gamma 𝑚 𝑎 𝑏 1 2 𝑚 1 {\displaystyle{\displaystyle\int_{\boldsymbol{\Omega}}|\mathbf{T}|^{c-\frac{1}% {2}(m+1)}\Psi\left(a;b;\mathbf{T}\right)\mathrm{d}\mathbf{T}=\frac{\Gamma_{m}% \left(c\right)\Gamma_{m}\left(a-c\right)\Gamma_{m}\left(c-b+\frac{1}{2}(m+1)% \right)}{\Gamma_{m}\left(a\right)\Gamma_{m}\left(a-b+\frac{1}{2}(m+1)\right)},}}
    0 references
    DLMF id
    DLMF:35.6.E8
    0 references
    constraint
    ℜ ⁡ ( a ) > ℜ ⁡ ( c ) + 1 2 ⁢ ( m - 1 ) > m - 1 𝑎 𝑐 1 2 𝑚 1 𝑚 1 {\displaystyle{\displaystyle\Re(a)>\Re(c)+\frac{1}{2}(m-1)>m-1}}
    0 references
    ℜ ⁡ ( c - b ) > - 1 𝑐 𝑏 1 {\displaystyle{\displaystyle\Re(c-b)>-1}}
    0 references
    Symbols used
    confluent hypergeometric function of matrix argument (second kind)
    Defining formula
    Ψ ⁡ ( a ; b ; 𝐓 ) confluent-hypergeometric-of-matrix-Psi 𝑎 𝑏 𝐓 {\displaystyle{\displaystyle\Psi\left(\NVar{a};\NVar{b};\NVar{\mathbf{T}}% \right)}}
    xml-id
    C35.S6.E2.m2aadec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1addec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3addec
    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.m1acdec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aedec
    0 references
    Q12726
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C35.S1.XMD1.m1fdec
    0 references
    Q12724
    Defining formula
    𝐓 𝐓 {\displaystyle{\displaystyle\mathbf{T}}}
    xml-id
    C35.S1.XMD11.m1gdec
    0 references
    Q12720
    Defining formula
    𝛀 𝛀 {\displaystyle{\displaystyle{\boldsymbol{\Omega}}}}
    xml-id
    C35.S1.XMD13.m1bdec
    0 references
    Q12728
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C35.S1.XMD2.m1fdec
    0 references
    Q12736
    Defining formula
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C35.S1.XMD3.m1dec
    0 references
    Q12723
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C35.S1.XMD8.m1edec
    0 references
     
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