# §35.5 Bessel Functions of Matrix Argument

## §35.5(i) Definitions

 35.5.1 $A_{\nu}\left(\boldsymbol{{0}}\right)=\frac{1}{\Gamma_{m}\left(\nu+\frac{1}{2}(% m+1)\right)},$ $\nu\in\mathbb{C}$.
 35.5.2 $A_{\nu}\left(\mathbf{T}\right)=A_{\nu}\left(\boldsymbol{{0}}\right)\sum_{k=0}^% {\infty}\frac{(-1)^{k}}{k!}\sum_{|\kappa|=k}\frac{1}{{\left[\nu+\frac{1}{2}(m+% 1)\right]_{\kappa}}}Z_{\kappa}\left(\mathbf{T}\right),$ $\nu\in\mathbb{C}$, $\mathbf{T}\in\boldsymbol{\mathcal{S}}$.
 35.5.3 $B_{\nu}\left(\mathbf{T}\right)=\int_{\boldsymbol{\Omega}}\operatorname{etr}% \left(-(\mathbf{T}\mathbf{X}+\mathbf{X}^{-1})\right)\left|\mathbf{X}\right|^{% \nu-\frac{1}{2}(m+1)}\,\mathrm{d}{\mathbf{X}},$ $\nu\in\mathbb{C}$, $\mathbf{T}\in{\boldsymbol{\Omega}}$. ⓘ Defines: $B_{\NVar{\nu}}\left(\NVar{\mathbf{T}}\right)$: Bessel function of matrix argument (second kind) Symbols: $\mathbb{C}$: complex plane, $\in$: element of, $\operatorname{etr}\left(\NVar{\mathbf{A}}\right)$: exponential of trace, $\int$: integral, $\mathbf{T}$: real symmetric $m\times m$ matrix, $\mathbf{X}$: real symmetric $m\times m$ matrix, $\left|\NVar{\mathbf{X}}\right|$: determinant of $\NVar{\mathbf{X}}$, $\,\mathrm{d}$: differential of matrix, ${\boldsymbol{\Omega}}$: space of positive-definite real symmetric $m\times m$ matrices and $m$: positive integer Referenced by: §35.9 Permalink: http://dlmf.nist.gov/35.5.E3 Encodings: TeX, pMML, png See also: Annotations for §35.5(i), §35.5 and Ch.35

## §35.5(ii) Properties

 35.5.4 $\int_{\boldsymbol{\Omega}}\operatorname{etr}\left(-\mathbf{T}\mathbf{X}\right)% \left|\mathbf{X}\right|^{\nu}A_{\nu}\left(\mathbf{S}\mathbf{X}\right)\,\mathrm% {d}{\mathbf{X}}=\operatorname{etr}\left(-\mathbf{S}\mathbf{T}^{-1}\right)\left% |\mathbf{T}\right|^{-\nu-\frac{1}{2}(m+1)},$ $\mathbf{S}\in\boldsymbol{\mathcal{S}}$, $\mathbf{T}\in{\boldsymbol{\Omega}}$; $\Re\left(\nu\right)>-1$.
 35.5.5 $\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{T}}A_{\nu_{1}}\left(\mathbf{S% }_{1}\mathbf{X}\right)\left|\mathbf{X}\right|^{\nu_{1}}\*A_{\nu_{2}}\left(% \mathbf{S}_{2}(\mathbf{T}-\mathbf{X})\right)\left|\mathbf{T}-\mathbf{X}\right|% ^{\nu_{2}}\,\mathrm{d}{\mathbf{X}}=\left|\mathbf{T}\right|^{\nu_{1}+\nu_{2}+% \frac{1}{2}(m+1)}A_{\nu_{1}+\nu_{2}+\frac{1}{2}(m+1)}\left((\mathbf{S}_{1}+% \mathbf{S}_{2})\mathbf{T}\right),$ $\nu_{j}\in\mathbb{C}$, $\Re\left(\nu_{j}\right)>-1$, $j=1,2$; $\mathbf{S}_{1},\mathbf{S}_{2}\in\boldsymbol{\mathcal{S}}$; $\mathbf{T}\in{\boldsymbol{\Omega}}$.
 35.5.6 $B_{\nu}\left(\mathbf{T}\right)=\left|\mathbf{T}\right|^{-\nu}B_{-\nu}\left(% \mathbf{T}\right),$ $\nu\in\mathbb{C}$, $\mathbf{T}\in{\boldsymbol{\Omega}}$.
 35.5.7 $\int_{\boldsymbol{\Omega}}A_{\nu_{1}}\left(\mathbf{T}\mathbf{X}\right)B_{-\nu_% {2}}\left(\mathbf{S}\mathbf{X}\right)\left|\mathbf{X}\right|^{\nu_{1}}\,% \mathrm{d}{\mathbf{X}}=\frac{1}{A_{\nu_{1}+\nu_{2}}\left(\boldsymbol{{0}}% \right)}\left|\mathbf{S}\right|^{\nu_{2}}\left|\mathbf{T}+\mathbf{S}\right|^{-% (\nu_{1}+\nu_{2}+\frac{1}{2}(m+1))},$ $\Re\left(\nu_{1}+\nu_{2}\right)>-1$; $\mathbf{S},\mathbf{T}\in{\boldsymbol{\Omega}}$.
 35.5.8 $\int_{\mathbf{O}(m)}\operatorname{etr}\left(\mathbf{S}\mathbf{H}\right)\mathrm% {d}{\mathbf{H}}=\frac{A_{-1/2}\left(-\frac{1}{4}\mathbf{S}\mathbf{S}^{\mathrm{% T}}\right)}{A_{-1/2}\left(\boldsymbol{{0}}\right)},$ $\mathbf{S}$ arbitrary.

## §35.5(iii) Asymptotic Approximations

For asymptotic approximations for Bessel functions of matrix argument, see Herz (1955) and Butler and Wood (2003).