# §35.3 Multivariate Gamma and Beta Functions

## §35.3(i) Definitions

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

 35.3.4 $\mathop{\Gamma_{m}\/}\nolimits\!\left(a\right)=\pi^{m(m-1)/4}\prod_{j=1}^{m}% \mathop{\Gamma\/}\nolimits\!\left(a-\tfrac{1}{2}(j-1)\right).$
 35.3.5 $\mathop{\Gamma_{m}\/}\nolimits\!\left(s_{1},\dots,s_{m}\right)=\pi^{m(m-1)/4}% \prod_{j=1}^{m}\mathop{\Gamma\/}\nolimits\!\left(s_{j}-\tfrac{1}{2}(j-1)\right).$
 35.3.6 $\mathop{\Gamma_{m}\/}\nolimits\!\left(a,\dots,a\right)=\mathop{\Gamma_{m}\/}% \nolimits\!\left(a\right).$ Symbols: $\mathop{\Gamma_{\NVar{m}}\/}\nolimits\!\left(\NVar{a}\right)$: 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)
 35.3.7 $\mathop{\mathrm{B}_{m}\/}\nolimits\!\left(a,b\right)=\frac{\mathop{\Gamma_{m}% \/}\nolimits\!\left(a\right)\mathop{\Gamma_{m}\/}\nolimits\!\left(b\right)}{% \mathop{\Gamma_{m}\/}\nolimits\!\left(a+b\right)}.$
 35.3.8 $\mathop{\mathrm{B}_{m}\/}\nolimits\!\left(a,b\right)=\int_{\boldsymbol{\Omega}% }|\mathbf{X}|^{a-\frac{1}{2}(m+1)}|\mathbf{I}+\mathbf{X}|^{-(a+b)}\mathrm{d}% \mathbf{X},$ $\Re{(a)},\Re{(b)}>\frac{1}{2}(m-1)$.