# §35.3 Multivariate Gamma and Beta Functions

## §35.3(i) Definitions

 35.3.1 $\Gamma_{m}\left(a\right)=\int_{\boldsymbol{\Omega}}\operatorname{etr}\left(-% \mathbf{X}\right)\left|\mathbf{X}\right|^{a-\frac{1}{2}(m+1)}\,\mathrm{d}{% \mathbf{X}},$ $\Re\left(a\right)>\frac{1}{2}(m-1)$.
 35.3.2 $\Gamma_{m}\left(s_{1},\dots,s_{m}\right)=\int_{\boldsymbol{\Omega}}% \operatorname{etr}\left(-\mathbf{X}\right)\left|\mathbf{X}\right|^{s_{m}-\frac% {1}{2}(m+1)}\prod_{j=1}^{m-1}|(\mathbf{X})_{j}|^{s_{j}-s_{j+1}}\,\mathrm{d}{% \mathbf{X}},$ $s_{j}\in\mathbb{C}$, $\Re\left(s_{j}\right)>\frac{1}{2}(j-1)$, $j=1,\dots,m$.
 35.3.3 $\mathrm{B}_{m}\left(a,b\right)=\int\limits_{\boldsymbol{{0}}<\mathbf{X}<% \mathbf{I}}\left|\mathbf{X}\right|^{a-\frac{1}{2}(m+1)}\left|\mathbf{I}-% \mathbf{X}\right|^{b-\frac{1}{2}(m+1)}\,\mathrm{d}{\mathbf{X}},$ $\Re\left(a\right),\Re\left(b\right)>\frac{1}{2}(m-1)$.

## §35.3(ii) Properties

 35.3.4 $\Gamma_{m}\left(a\right)=\pi^{m(m-1)/4}\prod_{j=1}^{m}\Gamma\left(a-\tfrac{1}{% 2}(j-1)\right).$
 35.3.5 $\Gamma_{m}\left(s_{1},\dots,s_{m}\right)=\pi^{m(m-1)/4}\prod_{j=1}^{m}\Gamma% \left(s_{j}-\tfrac{1}{2}(j-1)\right).$
 35.3.6 $\Gamma_{m}\left(a,\dots,a\right)=\Gamma_{m}\left(a\right).$ ⓘ Symbols: $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$: multivariate gamma function, $a$: complex variable and $m$: positive integer Permalink: http://dlmf.nist.gov/35.3.E6 Encodings: TeX, pMML, png See also: Annotations for §35.3(ii), §35.3 and Ch.35
 35.3.7 $\mathrm{B}_{m}\left(a,b\right)=\frac{\Gamma_{m}\left(a\right)\Gamma_{m}\left(b% \right)}{\Gamma_{m}\left(a+b\right)}.$
 35.3.8 $\mathrm{B}_{m}\left(a,b\right)=\int_{\boldsymbol{\Omega}}\left|\mathbf{X}% \right|^{a-\frac{1}{2}(m+1)}\left|\mathbf{I}+\mathbf{X}\right|^{-(a+b)}\,% \mathrm{d}{\mathbf{X}},$ $\Re\left(a\right),\Re\left(b\right)>\frac{1}{2}(m-1)$.