## 2 Spectral Functional Equations in an Angular Region Filled by Arbitrary Linear Media
Repeating the same procedure for region 2 in Figure 1a, we obtain the functional equations as the solution of the differential Equation (19) in the Laplace domain using the left Laplace transform \(\mathbf{\tilde{\psi}_{y}}(\eta,v)\doteq\int\limits_{-\infty}^{0}e^{j\eta u}\mathbf{ \psi_{y}}(u,v)du\):  
\[-\frac{d}{d\mathbf{\tilde{\psi}_{y}}}(\eta,v)=\mathbf{M_{\gamma}}(-\mu\alpha_{o},-j\eta )\cdot\mathbf{\tilde{\psi}_{y}}(\eta,v)+\mathbf{\psi_{sb}}(v),\ v<0 \tag{36}\]  
where \(\mathbf{\psi_{sb}}(v)\) has the same expression of \(\mathbf{\psi_{sa}}(v)\) (27) but with a different support \(v<0\), and allows the incorporation of boundary conditions along face \(b\) (\(u=0_{+},v<0\)). The application of the characteristic Green function procedure yields expression (28) for region 2 in Figure 1a, which is identical to that for region 1 except for \(C_{i}\) and the source term \(\mathbf{\psi_{sb}}(v)\), which depend on the local constitutive parameters and boundary conditions of region 2. Now, considering the asymptotic behavior of the exponential function in \(v\) for \(v\to-\infty\), we need to have \(C_{1}=C_{2}=0\); at the same time, the second couple of integrals are null. For this reason, setting \(v=0\), we obtain  
\[\tilde{\psi}_{\phi+}(\eta)\doteq\tilde{\psi}_{y}(\eta,0)=C_{3}u_{3}+C_{4}u_{4}- \sum\limits_{i=1}^{2}u_{i}v_{i}\cdot\int\limits_{0}^{\infty}e^{-\lambda_{i}( \gamma)(v-v^{\prime})}\psi_{sb}(v^{\prime})dv^{\prime}. \tag{37}\]  
By weighting (37) with the reciprocal vectors \(v_{1},v_{2}\) of \(M_{\gamma}\), we obtain the functional equations  
\[v_{i}\cdot\tilde{\psi}_{\phi+}(\eta)=-v_{i}\cdot\tilde{\psi}_{sb+}(-m_{i}( \gamma)),\quad i=1,2, \tag{38}\]  
where we have used the definition of the \(v\) left Laplace transform  
\[\tilde{\psi}_{sb+}(-m_{i}(\gamma))\doteq\int\limits_{-\infty}^{0}e^{-jm_{i}( \gamma)^{2}}\psi_{sb}(v)dv=\int\limits_{0}^{\infty}e^{-jm_{i}(\gamma)\rho} \psi_{sb}(-\rho)d\rho. \tag{39}\]  
Note the differences and similarities between Laplace transformations (31) and (39), which yield the same definition of the \(-m_{i}(\gamma)\) Laplace transform in \(\rho\) but are applied to different quantities. Furthermore, the regularity properties of the \(-m_{i}(\gamma)\) Laplace transform are inherited from \(\xi_{i}\left(Im[\xi_{i}]<0\right)\) according to (24)-(25). With reference to Figure 1a and its caption, analyzing and expanding \(\tilde{\psi}_{sb+}(-m_{i}(\gamma))\) in (38), we rephrase the functional equations into  
\[v_{i}\cdot\tilde{\psi}_{\phi+}(\eta)=-v_{i}\cdot T(\gamma)\cdot\mathbf{P}\cdot \tilde{\psi}_{b+}(-m_{i}(\gamma),-\pi+\gamma),\quad i=1,2. \tag{40}\]  
In (40), \(T(\gamma)\) is again (35); as for region \(1\), \(\mathbf{P}=diag\{1,-1,1,-1\}\) is needed for \(v=-X_{2}\) in region \(2\) with respect to \(v=X_{1}\) in region \(1\), \(\tilde{\psi}_{\phi+}(\eta)\) is the \(\eta\) Laplace transform of the tangent-to-face-\(o\) field components reported in (33), and \(\tilde{\psi}_{b+}(-m_{i}(\gamma),-\pi+\gamma)\) is the \(-m_{i}(\gamma)\) Laplace transform of the tangent-to-face-\(b\) field components (i.e., at \(\varphi=-\pi+\gamma\)) in the local-to-face-\(b\) Cartesian \((z,X_{2},Y_{2})\) coordinates and global cylindrical coordinates \((z,\rho,\varphi)\) of Figure 1a. \[ \begin{array}{ll}\tilde{\psi}_{b+}(-m_{i}(\gamma),-\pi+\gamma)&=\int\limits _{0}^{\infty}|E_{z}(X_{2},0),E_{X_{2}}(X_{2},0),H_{z}(X_{2},0),H_{X_{2}}(X_{2 },0)|^{t}e^{-jm_{i}(\gamma)X_{2}}dx\\ &=\int\limits_{0}^{\infty}|E_{z}(\rho,-\pi+\gamma),E_{\rho}(\rho,-\pi+\gamma),H_{z}(\rho,-\pi+\gamma),H_{\rho}(\rho,-\pi+\gamma)|^{t}e^{-jm_{i}(\gamma) \rho}d\rho\end{array}  \tag{41}\]  
While considering the wedge scattering problem with symmetry with respect to the \(x\) axis, in combination with region \(1\) in Figure 1a, we need to consider region \(2^{\prime}\) in Figure 1b, where \(\gamma\to\pi-\gamma\) with respect to region \(2\) in Figure 1a, i.e., for the same face \(a\), we change the orientation of face \(b\) at \(\varphi=\gamma\) from \(\varphi=-\pi+\gamma\) to \(\varphi=-\gamma\).