## 2 Spectral Functional Equations in an Angular Region Filled by Arbitrary Linear Media
The functional equations of region \(2^{\prime}\) become  
\[v_{i}\cdot\tilde{\psi}_{\phi+}(\eta)=-v_{i}\cdot T(\pi-\gamma)\cdot\mathbf{P} \cdot\tilde{\psi}_{b+}(-m_{i}(\pi-\gamma),-\gamma),\quad i=1,2, \tag{42}\]  
where  
\[ \begin{array}{ll}\tilde{\psi}_{b+}(-m_{i}(\pi-\gamma),-\gamma)&=\int\limits _{0}^{\infty}|E_{z}(X_{2},0),E_{X_{2}}(X_{2},0),H_{z}(X_{2},0),H_{X_{2}}(X_{2 },0)|^{t}e^{-jm_{i}(\pi-\gamma)X_{2}}dx\\ &=\int\limits_{0}^{\infty}|E_{z}(\rho,-\gamma),E_{\rho}(\rho,-\gamma),H_{z}( \rho,-\gamma),H_{\rho}(\rho,-\gamma)|^{t}e^{-jm_{i}(\pi-\gamma)\rho}d\rho, \end{array}  \tag{43}\]  
which is the \(-m_{i}(\pi-\gamma)\) Laplace transform of the tangent-to-face-\(b\) field components (i.e., now at \(\varphi=-\gamma\)) in local-to-face-\(b\) Cartesian \((z,X_{2},Y_{2})\) coordinates and global cylindrical coordinates \((z,\rho,\varphi)\) in Figure 1b. Note that in (42) we assumed that region \(2^{\prime}\) is homogeneous to region \(1\), yielding the same \(u_{i},v_{i}\); otherwise, specific vectors would be needed. Equations (42) are functional equations that relate the Laplace transforms of combinationsof field components on the boundaries of the angular region \(2^{\prime}\) in Figure 1b, i.e., face \(o\)\(u>0\), \(v=0\) (\(\varphi=0\)) and face \(b\)\(u=0\), \(v<0\) (\(\varphi=-\gamma\)). In (42), note the new dependence of \(\mathbf{T}(\cdot)\) (35) on \(\pi-\gamma\) due to the effect of anisotropies while changing the orientation of face \(b\) from \(-\pi+\gamma\) to \(-\gamma\). Furthermore, in the case of symmetric media (\(\lambda_{1,2}=-\lambda_{3,4}\)), we have \(m_{3,4}(\gamma)=m_{1,2}(\pi-\gamma)\); see (24)-(25). In general, the system of functional equations in (32) and (42) allows for the analysis of angular regions that are symmetric with respect to the \(x\) axis; these are at the base of the analysis of scattering problems constituted by impenetrable and penetrable wedges surrounded/made by arbitrary linear media. In the following, to investigate a practical scattering problem, we impose boundary conditions at the faces of each angular region on the functional Equations (32) and (42), yielding a system of GWHEs. From Functional Equations to GWHEs and Their Regularized Integral Representations with a Network Interpretation  
Network representations of angular regions in isotropic media for electromagnetic scattering were extensively studied in multiple spectral domains in [13] using algebraic and integral formalisms. The proposed equations were effectively applied in several works to practical wedge scattering problems; see [11; 12] and the references therein. Furthermore, network formalisms have been effectively applied for complex canonical problems containing angular and layered regions in isotropic media; see for instance the double wedge [31], flanged dielectric loaded waveguide [32], and wedge over dielectric layer [30], among others. In arbitrary linear media, the system of functional Equations (32) and (42),  
\[ \begin{array}{l}\mathbf{v}_{i}\cdot\mathbf{\tilde{\psi}}_{o+}(\eta)=\mathbf{v}_{i}\cdot \mathbf{T}(\gamma)\cdot\mathbf{\tilde{\psi}}_{a+}(-m_{i}(\gamma),\gamma),\quad i=3,4 \\ \mathbf{v}_{i}\cdot\mathbf{\tilde{\psi}}_{o+}(\eta)=-\mathbf{v}_{i}\cdot\mathbf{T}(\pi-\gamma) \cdot\mathbf{P}\cdot\mathbf{\tilde{\psi}}_{b+}(-m_{i}(\pi-\gamma),-\gamma),\quad i=1,2 \end{array}  \tag{44}\]  
constitutes two systems of network relations that link the spectral field components in region 1 and region \(2^{\prime}\) (Figure 1), respectively, via a sort of two-port transmission relations in algebraic form. Looking at the first system in (44), we have two combinations of \(\mathbf{\tilde{\psi}}_{o+}(\eta)\) components (33) related to two combinations of \(\mathbf{\tilde{\psi}}_{a+}(-m_{i}(\gamma),\gamma)\) components (34), i.e., with reference to Figure 1a, the tangential field components of face \(o\) are related to tangential field components of face \(a\).