# §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 $B_{\nu}\left(\mathbf{T}\right)=\int_{\boldsymbol{\Omega}}\mathrm{etr}\left(-(% \mathbf{T}\mathbf{X}+\mathbf{X}^{-1})\right)|\mathbf{X}|^{\nu-\frac{1}{2}(m+1)% }\mathrm{d}\mathbf{X},$ $\nu\in\mathbb{C}$, $\mathbf{T}\in{\boldsymbol{\Omega}}$. ⓘ Defines: $B_{\NVar{\nu}}\left(\NVar{\mathbf{T}}\right)$: Bessel function of matrix argument (second kind) Symbols: $\mathbb{C}$: complex plane, $\mathrm{d}\NVar{x}$: differential of $x$, $\in$: element of, $\mathrm{etr}\left(\NVar{\mathbf{X}}\right)$: exponential of trace, $\int$: integral, ${\boldsymbol{\Omega}}$: space of 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 35

## §35.5(ii) Properties

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

## §35.5(iii) Asymptotic Approximations

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