Formula:10498

Digital Library of Mathematical Functions ID 35.6.E5

${\displaystyle{\displaystyle\int_{\boldsymbol{\Omega}}\mathrm{etr}\left(-\mathbf{T}\mathbf{X}\right)|\mathbf{X}|^{b-\frac{1}{2}(m+1)}{{}_{1}F_{1}}\left({a\atop b};\mathbf{S}\mathbf{X}\right)\mathrm{d}\mathbf{X}=\Gamma_{m}\left(b\right)|\mathbf{I}-\mathbf{S}\mathbf{T}^{-1}|^{-a}|\mathbf{T}|^{-b},}}$

Constraint(s)

${\displaystyle{\displaystyle\Re(b)>\frac{1}{2}(m-1)}}$
${\displaystyle{\displaystyle\mathbf{T}>\mathbf{S}}}$
${\displaystyle{\displaystyle\Re(b)>\frac{1}{2}(m-1)}}$

Symbols List

• ${\displaystyle{\displaystyle{{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_{q}};\NVar{\mathbf{T}}\right)}}$: generalized hypergeometric function of matrix argument
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\mathrm{etr}\left(\NVar{\mathbf{X}}\right)}}$:
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle\Gamma_{\NVar{m}}\left(\NVar{a}\right)}}$:
• ${\displaystyle{\displaystyle\Re}}$: real part
• ${\displaystyle{\displaystyle a}}$: complex variable
• ${\displaystyle{\displaystyle\mathbf{S}}}$: identity matrix
• ${\displaystyle{\displaystyle\mathbf{T}}}$: space of real symmetric matrices
• ${\displaystyle{\displaystyle\mathbf{X}}}$: real symmetric matrix
• ${\displaystyle{\displaystyle{\boldsymbol{\Omega}}}}$: real symmetric matrix
• ${\displaystyle{\displaystyle b}}$: complex variable
• ${\displaystyle{\displaystyle m}}$: positive integer
 

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