Fundamental matrices of homogeneous hyperbolic systems. Applications to crystal optics, elastodynamics, and piezoelectromagnetism

Modifying L. Gårding's derivation in the scalar case we deduce Herglotz‐Petrovsky formulae for fundamental matrices (“Green's tensors”) for real homogeneous hyperbolic systems of partial differential operators. As an application, we calculate the fundamental matrix for elastodynamic systems of hexagonal symmetry with reducible determinant (Props. 1, 2). A special case thereof is the fundamental matrix of the system of uniaxial optics (Prop. 3). The calculations are based on integrals of the type $\int_C ([\xi,p]/[\xi,q]) \delta([\xi,x]) \hbox{d}o(\xi)$ where the 1904. conic section C in \documentclass{article}\pagestyle{empty}\usepackage{amssymb}\begin{document}$\mathbb{R}^4$\end{document} is defined as the intersection of [ξ,ξ] = 0 with [ξ, N ] = 1.