## 2 Spectral Functional Equations in an Angular Region Filled by Arbitrary Linear Media
For this reason, setting \(v=0\), we obtain the following spectral field representation along face \(o\):  
\[-\frac{\partial}{\partial v}\psi_{\mathbf{y}}(u,v)=M_{\gamma}(-j\alpha_{o},\frac {\partial}{\partial u})\cdot\mathbf{\psi}_{\mathbf{y}}(u,v). \tag{19}\]2  
By weighting (29) with the reciprocal vectors \(\mathbf{v_{3}},\mathbf{v_{4}}\) of \(\mathbf{M_{\gamma}}\), we obtain the following functional equations:  
\[-\frac{\partial}{\partial v}\psi_{\mathbf{y}}(u,v)=M_{\gamma}(-j\alpha_{o},\frac {\partial}{\partial u})\cdot\mathbf{\psi}_{\mathbf{y}}(u,v). \tag{19}\]3where we use the following definition of the Laplace transform:  
\[-\frac{\partial}{\partial v}\psi_{\mathbf{y}}(u,v)=M_{\gamma}(-j\alpha_{o},\frac {\partial}{\partial u})\cdot\mathbf{\psi}_{\mathbf{y}}(u,v). \tag{19}\]4  
With reference to Figure 1a and its caption, analyzing and expanding \(\mathbf{\tilde{\psi}_{sa+}}(-m_{i}(\gamma))\) in (30) using [PERSON]'s equations, we rephrase the functional equations for region 1 as  
\[-\frac{\partial}{\partial v}\psi_{\mathbf{y}}(u,v)=M_{\gamma}(-j\alpha_{o},\frac {\partial}{\partial u})\cdot\mathbf{\psi}_{\mathbf{y}}(u,v). \tag{19}\]5  
where \(\mathbf{\tilde{\psi}_{o+}}(\eta)\) is the \(\eta\) Laplace transform of the tangent-to-face-\(o\) field components (i.e., at \(\varphi=0\)) in Cartesian \((z,x,y)\) and cylindrical coordinates \((z,\rho,\varphi)\) (omitting the \(z\) coordinate for invariance)  
\[-\frac{\partial}{\partial v}\psi_{\mathbf{y}}(u,v)=M_{\gamma}(-j\alpha_{o},\frac {\partial}{\partial u})\cdot\mathbf{\psi}_{\mathbf{y}}(u,v). \tag{19}\]6 \[-\frac{\partial}{\partial v}\psi_{\mathbf{y}}(u,v)=M_{\gamma}(-j\alpha_{o},\frac {\partial}{\partial u})\cdot\mathbf{\psi}_{\mathbf{y}}(u,v). \tag{19}\]7  
where \(\mathbf{\tilde{\psi}_{a+}}(-m_{i}(\gamma),\gamma)\) is the \(-m_{i}(\gamma)\) Laplace transform of the tangent-to-face-\(a\) field components (i.e., at \(\varphi=\gamma\)) in local-to-face-\(a\) Cartesian \((z,X_{1},Y_{1})\) coordinates and global cylindrical coordinates \((z,\rho,\varphi)\) (located at \(\varphi=+\gamma\))  
\[-\frac{\partial}{\partial v}\psi_{\mathbf{y}}(u,v)=M_{\gamma}(-j\alpha_{o},\frac {\partial}{\partial u})\cdot\mathbf{\psi}_{\mathbf{y}}(u,v). \tag{19}\]8 \[-\frac{\partial}{\partial v}\psi_{\mathbf{y}}(u,v)=M_{\gamma}(-j\alpha_{o},\frac {\partial}{\partial u})\cdot\mathbf{\psi}_{\mathbf{y}}(u,v). \tag{19}\]9  
and  
\[\mathbf{T}(\gamma)=\left( \begin{array}{ccc}\frac{\sin(\gamma)(\alpha_{o}\xi_{ yy}+\xi_{xy}\xi_{yz}\xi_{yz}-\mu_{xy}\omega\xi_{xy})}{\omega(\mu_{yy}\xi_{yz}-\xi_{yy} \xi_{yy})}+\cos(\gamma)&0&\frac{\sin(\gamma)(\alpha_{o}\mu_{yy}+\xi_{xy}\mu_{ yy}\mu-\xi_{yy}\mu_{xy}\omega)}{\omega(\mu_{yy}\xi_{yz}-\xi_{yy}\xi_{yy})}&0\\ \frac{\sin(\gamma)(-\xi_{xy}\xi_{yz}\xi_{yz}+\xi_{yz}\xi_{yz}+\mu_{z}\omega\xi_{ xy})}{\omega(\mu_{yy}\xi_{yz}-\xi_{yy}\xi_{yy})}&1&\frac{\sin(\gamma)(\xi_{yz} \xi_{yz}\xi_{yz}\xi_{yz}-\xi_{yy}\xi_{yz}+\mu_{yy}\xi_{yy})}{\omega(\mu_{yy} \xi_{yz}-\xi_{yy}\xi_{yy})}&0\\ \frac{\sin(\gamma)(-\alpha_{o}\xi_{yy}-\xi_{yy}\xi_{xx}+\xi_{yz}\alpha\omega_{ xy})}{\omega(\mu_{yy}\xi_{yz}-\xi_{yy}\xi_{yy})}&0&\cos(\gamma)-\frac{\sin(\gamma)( \alpha_{o}\xi_{yy}-\xi_{yy}\xi_{xy}\omega+\mu_{yy}\alpha\xi_{xy})}{\omega(\mu_{ yy}\xi_{yz}-\xi_{yy}\xi_{yy})}&0\\ \frac{\sin(\gamma)(\xi_{yz}\alpha_{xz}-\xi_{yy}(\eta+\xi_{z}\alpha\omega))}{ \omega(\mu_{yy}\xi_{yz}-\xi_{yy}\xi_{yy})}&0&\frac{\sin(\gamma)(\mu_{yy}\alpha \omega_{xy}-\xi_{yy}(\eta+\xi_{z}\alpha\omega))}{\omega(\mu_{yy}\xi_{yz}-\xi_{ yy}\xi_{yy})}&1\end{array} \right) \tag{35}\]  
Note that (32) are functional equations that relate the Laplace transforms of combinations of field components on the boundaries of angular region 1 in Figure 1a, i.e., face \(o\)\(u>0\), \(v=0\) (\(\varphi=0\)) and face \(a\)\(u=0\), \(v>0\) (\(\varphi=\gamma\)). Furthermore, we observe that the angle \(\gamma\) is essential in determining the impact of anisotropies through \(\mathbf{T}(\gamma)\).