# §35.3 Multivariate Gamma and Beta Functions

## §35.3(i) Definitions

 35.3.1 $\Gamma_{m}\left(a\right)=\int_{\boldsymbol{\Omega}}\mathrm{etr}\left(-\mathbf{% X}\right)|\mathbf{X}|^{a-\frac{1}{2}(m+1)}\mathrm{d}\mathbf{X},$ $\Re(a)>\frac{1}{2}(m-1)$.
 35.3.2 $\Gamma_{m}\left(s_{1},\dots,s_{m}\right)=\int_{\boldsymbol{\Omega}}\mathrm{etr% }\left(-\mathbf{X}\right)|\mathbf{X}|^{s_{m}-\frac{1}{2}(m+1)}\prod_{j=1}^{m-1% }|(\mathbf{X})_{j}|^{s_{j}-s_{j+1}}\mathrm{d}\mathbf{X},$ $s_{j}\in\mathbb{C}$, $\Re(s_{j})>\frac{1}{2}(j-1)$, $j=1,\dots,m$.
 35.3.3 $\mathrm{B}_{m}\left(a,b\right)=\int\limits_{\boldsymbol{{0}}<\mathbf{X}<% \mathbf{I}}|\mathbf{X}|^{a-\frac{1}{2}(m+1)}|\mathbf{I}-\mathbf{X}|^{b-\frac{1% }{2}(m+1)}\mathrm{d}\mathbf{X},$ $\Re(a),\Re(b)>\frac{1}{2}(m-1)$. ⓘ Defines: $\mathrm{B}_{\NVar{m}}\left(\NVar{a},\NVar{b}\right)$: multivariate beta function Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\Re$: real part, $a$: complex variable, $b$: complex variable and $m$: positive integer Keywords: multivariate beta function Permalink: http://dlmf.nist.gov/35.3.E3 Encodings: TeX, pMML, png See also: Annotations for 35.3(i), 35.3 and 35

## §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 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 $\mathrm{B}_{m}\left(a,b\right)=\int_{\boldsymbol{\Omega}}|\mathbf{X}|^{a-\frac% {1}{2}(m+1)}|\mathbf{I}+\mathbf{X}|^{-(a+b)}\mathrm{d}\mathbf{X},$ $\Re(a),\Re(b)>\frac{1}{2}(m-1)$.