# §35.5 Bessel Functions of Matrix Argument

## §35.5(i) Definitions

 35.5.1 $\mathop{A_{\nu}\/}\nolimits\!\left(\boldsymbol{{0}}\right)=\frac{1}{\mathop{% \Gamma_{m}\/}\nolimits\!\left(\nu+\frac{1}{2}(m+1)\right)},$ $\nu\in\mathbb{C}$.
 35.5.2 $\mathop{A_{\nu}\/}\nolimits\!\left(\mathbf{T}\right)=\mathop{A_{\nu}\/}% \nolimits\!\left(\boldsymbol{{0}}\right)\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!}% \sum_{|\kappa|=k}\frac{1}{{\left[\nu+\frac{1}{2}(m+1)\right]_{\kappa}}}\mathop% {Z_{\kappa}\/}\nolimits\!\left(\mathbf{T}\right),$ $\nu\in\mathbb{C}$, $\mathbf{T}\in\boldsymbol{\mathcal{S}}$.
 35.5.3 $\mathop{B_{\nu}\/}\nolimits\!\left(\mathbf{T}\right)=\int_{\boldsymbol{\Omega}% }\mathop{\mathrm{etr}\/}\nolimits\!\left(-(\mathbf{T}\mathbf{X}+\mathbf{X}^{-1% })\right)|\mathbf{X}|^{\nu-\frac{1}{2}(m+1)}\mathrm{d}\mathbf{X},$ $\nu\in\mathbb{C}$, $\mathbf{T}\in{\boldsymbol{\Omega}}$. Defines: $\mathop{B_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{\mathbf{T}}\right)$: Bessel function of matrix argument (second kind) Symbols: $\mathbb{C}$: complex plane, $\mathrm{d}\NVar{x}$: differential of $x$, $\in$: element of, $\mathop{\mathrm{etr}\/}\nolimits\!\left(\NVar{\mathbf{X}}\right)$: 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(ii) Properties

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

## §35.5(iii) Asymptotic Approximations

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