§19.31 Probability Distributions

$R_{G}\left(x,y,z\right)$ and $R_{F}\left(x,y,z\right)$ occur as the expectation values, relative to a normal probability distribution in ${\mathbb{R}^{2}}$ or ${\mathbb{R}^{3}}$, of the square root or reciprocal square root of a quadratic form. More generally, let $\mathbf{A}$ ($=[a_{r,s}]$) and $\mathbf{B}$ ($=[b_{r,s}]$) be real positive-definite matrices with $n$ rows and $n$ columns, and let $\lambda_{1},\dots,\lambda_{n}$ be the eigenvalues of $\mathbf{A}\mathbf{B}^{-1}$. If $\mathbf{x}$ is a column vector with elements $x_{1},x_{2},\dots,x_{n}$ and transpose $\mathbf{x}^{\mathrm{T}}$, then

 19.31.1 $\mathbf{x}^{\mathrm{T}}\mathbf{A}\mathbf{x}=\sum_{r=1}^{n}\sum_{s=1}^{n}a_{r,s% }x_{r}x_{s},$ ⓘ Symbols: $n$: nonnegative integer Permalink: http://dlmf.nist.gov/19.31.E1 Encodings: TeX, pMML, png See also: Annotations for 19.31 and 19

and

 19.31.2 $\int_{{\mathbb{R}^{n}}}(\mathbf{x}^{\mathrm{T}}\mathbf{A}\mathbf{x})^{\mu}\exp% \left(-\mathbf{x}^{\mathrm{T}}\mathbf{B}\mathbf{x}\right)\mathrm{d}x_{1}\cdots% \mathrm{d}x_{n}=\frac{\pi^{n/2}\Gamma\left(\mu+\tfrac{1}{2}n\right)}{\sqrt{% \det\mathbf{B}}\Gamma\left(\tfrac{1}{2}n\right)}R_{\mu}\left(\tfrac{1}{2},% \dots,\tfrac{1}{2};\lambda_{1},\dots,\lambda_{n}\right),$ $\mu>-\tfrac{1}{2}n$.

§19.16(iii) shows that for $n=3$ the incomplete cases of $R_{F}$ and $R_{G}$ occur when $\mu=-1/2$ and $\mu=1/2$, respectively, while their complete cases occur when $n=2$.

For (19.31.2) and generalizations see Carlson (1972b).