## 2 Spectral Functional Equations in an Angular Region Filled by Arbitrary Linear Media
The same rationale is applied to region 2 to double the equations of that region (second line of (44), as also reported in (47)) by obtaining  
\[ \begin{array}{l}\mathbf{v}_{\mathbf{\Omega}_{2}}\cdot\mathbf{\tilde{\Psi}_{\mathbf{b}+}}(\eta)= \mathbf{v}_{\mathbf{\Omega}_{2}}\cdot\mathbf{T}_{\mathbf{\Omega}_{2}}(\gamma)\cdot\mathbf{\tilde{ \Psi}_{\mathbf{o}+}}(-m_{iY_{2}}(\gamma),\gamma),\quad i=3,4,\\ \mathbf{v}_{i}\cdot\mathbf{\tilde{\Psi}_{\mathbf{o}+}}(\eta)=-\mathbf{v}_{i}\cdot\mathbf{T}(\pi- \gamma)\cdot\mathbf{P}\cdot\mathbf{\tilde{\Psi}_{\mathbf{b}+}}(-m_{i}(\pi-\gamma),-\gamma),\quad i=1,2.\end{array}  \tag{47}\]  
In the first pair of equations in (47), we have used the subscript \(Y_{2}\) to make reference to a rotated coordinated system \((z,X_{2},Y_{2})\) with respect to \((z,x,y)\); see region 2 in Figure 2c and the related region \(1^{\prime}\) in Figure 2d. We note that the first pair of equations in (47) can be easily derived by studying the classical region 1 (see the first pair of equations in (44)) except with modified definitions of \(\mathbf{v}_{\mathbf{\Omega}_{2}}\), \(\mathbf{T}_{\mathbf{\Omega}_{2}}(\gamma)\), and \(m_{iY_{2}}(\gamma)\) because of their dependence on the constitutive parameters \((\underline{\underline{\mathbf{\varepsilon}}},\underline{\underline{\mathbf{\mu}}}, \underline{\underline{\mathbf{\varepsilon}}},\underline{\underline{\mathbf{\varepsilon }}})\), which are redefined in the reference coordinate system \((z,X_{2},Y_{2})\), i.e., \((\underline{\underline{\mathbf{\varepsilon}}}_{Y_{2}},\underline{\underline{\mathbf{ \mu}}}_{Y_{2}},\underline{\underline{\mathbf{\varepsilon}}}_{Y_{2}},\underline{ \underline{\mathbf{\varepsilon}}}_{Y_{2}})\), for example,  
\[\underline{\underline{\mathbf{\varepsilon}}}_{Y_{2}}=\underline{\underline{\mathbf{R }}}_{Y_{2}}^{-1}\cdot\underline{\underline{\mathbf{\varepsilon}}}\cdot\underline{ \underline{\mathbf{R}}}_{Y_{2}},\quad\underline{\underline{\mathbf{R}}}_{Y_{2}}= \left( \begin{array}{ccc}\cos(\gamma)&\sin(\gamma)&0\\ -\sin(\gamma)&\cos(\gamma)&0\\ 0&0&1\end{array} \right), \tag{48}\]  
due to the rotation of \(-\gamma\). The sets of Equations (45) and (47) constitute a complete set of functional equations that respectively describe regions 1 and 2 in Figure 3. In the case of symmetric media (i.e., \(\lambda_{1,2}=-\lambda_{3,4}\)) we have \(m_{3,4}(\gamma)=m_{1,2}(\pi-\gamma)\); see (24)-(25). In isotropic media, it is always possible to introduce the angular complex plane \(w\) and the KL transform method [13], where functional equations become two-port admittance relations of the Norton type in integral and algebraic forms in a unique complex plane. In arbitrary linear media, the definition of such complex planes is not possible; however, a novel method that resorts to the following Cauchy decomposition formula in the \(-m(\eta)\) plane is introduced. This is a fundamental tool that allows for the description of angular region problems in arbitrary linear media without introducing further complex planes other than the initial Laplace transforms. In particular, to obtain regularized integral equations from GWHEs, it is not necessary to map the GWHEs into CWHEs with suitable  
Figure 3: Two angular regions that are symmetric with respect to the x axis of the aperture angle \(\gamma\) and represent wedge problems immersed in arbitrary linear media; they are modeled using the complete sets of Equations in (45) and (47). transformations before the application of Fredholm factorization (originally ideated and valid only for the CWHEs). This revisited novel version of the regularization procedure can be called the _direct Fredholm factorization method_.