## 2 Spectral Functional Equations in an Angular Region Filled by Arbitrary Linear Media
In any case, the types of completed functional equations and constraints with boundary conditions are always of the same form and are a well-posed mathematical problem of GWHEs. As a simple example to illustrate the procedure, let us consider a problem constituted by only region 1 with PEC boundary conditions when filled by arbitrary linear media. In this case, we obtain  
\[ \begin{split} A_{11}(\eta)H_{oz}(\eta)+A_{12}(\eta)H_{op}(\eta)& =B_{11}(\eta)H_{az}(-m_{3})+B_{12}(\eta)H_{ap}(-m_{3})\\ A_{21}(\eta)H_{oz}(\eta)+A_{22}(\eta)H_{op}(\eta)& =B_{21}(\eta)H_{az}(-m_{4})+B_{22}(\eta)H_{ap}(-m_{4})\\ A_{31}(\eta)H_{az}(\eta)+A_{32}(\eta)H_{ap}(\eta)& =B_{31}(\eta)H_{oz}(-m_{1})+B_{32}(\eta)H_{op}(-m_{1})\\ A_{41}(\eta)H_{az}(\eta)+A_{42}(\eta)H_{ap}(\eta)& =B_{41}(\eta)H_{oz}(-m_{2})+B_{42}(\eta)H_{op}(-m_{2})\end{split}  \tag{51}\]  
where on the LHS we have plus field unknowns in \(\eta\) and on the RHS we have minus field unknowns in \(m_{i}()\). The apparent redundancy in (51) after the imposition of boundary conditions is exploited to obtain integral representations only in terms of the field components \(H_{oz}(\eta)\), \(H_{op}(\eta)\), \(H_{az}(\eta)\), \(H_{ap}(\eta)\) in the unique complex plane \(\eta\) using (49). Furthermore, the application of the novel version of the Fredholm factorization method allows us to obtain regularized integral equations. We assert that this procedure is applicable to GWHEs in general, not only for the specific problem presented in this simple example. The application of (49) to the RHS of (51) yields the following:  
\[ \begin{split} A_{11}(\eta)H_{oz}(\eta)+A_{12}(\eta)H_{op}(\eta)& =\tfrac{B_{11}(\eta)}{2\pi)}\int\limits_{-\infty}^{\infty}\tfrac{ H_{az}(\eta^{\prime})}{\eta^{\prime}+m_{3}}d\eta^{\prime}+\tfrac{B_{12}(\eta)}{2 \pi)}\int\limits_{-\infty}^{\infty}\tfrac{H_{ap}(\eta^{\prime})}{\eta^{\prime} +m_{3}}d\eta^{\prime}+H_{az}^{ns}(-m_{3})+H_{ap}^{ns}(-m_{3})\\ A_{21}(\eta)H_{oz}(\eta)+A_{22}(\eta)H_{op}(\eta)& =\tfrac{B_{21}(\eta)}{2\pi)}\int\limits_{-\infty}^{\infty}\tfrac{ H_{az}(\eta^{\prime})}{\eta^{\prime}+m_{4}}d\eta^{\prime}+\tfrac{B_{22}(\eta)}{2 \pi)}\int\limits_{-\infty}^{\infty}\tfrac{H_{ap}(\eta^{\prime})}{\eta^{\prime} +m_{4}}d\eta^{\prime}+H_{az}^{ns}(-m_{4})+H_{ap}^{ns}(-m_{4})\\ A_{31}(\eta)H_{az}(\eta)+A_{32}(\eta)H_{ap}(\eta)& =\tfrac{B_{31}(\eta)}{2\pi)}\int\limits_{-\infty}^{\infty}\tfrac{ H_{az}(\eta^{\prime})}{\eta^{\prime}+m_{1}}d\eta^{\prime}+\tfrac{B_{22}(\eta)}{2 \pi)}\int\limits_{-\infty}^{\infty}\tfrac{H_{ap}(\eta^{\prime})}{\eta^{\prime} +m_{1}}d\eta^{\prime}+H_{oz}^{ns}(-m_{1})+H_{op}^{ns}(-m_{1})\\ A_{41}(\eta)H_{az}(\eta)+A_{42}(\eta)H_{ap}(\eta)& =\tfrac{B_{41}(\eta)}{2\pi)}\int\limits_{-\infty}^{\infty}\tfrac{ H_{az}(\eta^{\prime})}{\eta^{\prime}+m_{2}}d\eta^{\prime}+\tfrac{B_{22}(\eta)}{2 \pi)}\int\limits_{-\infty}^{\infty}\tfrac{H_{ap}(\eta^{\prime})}{\eta^{\prime} +m_{2}}d\eta^{\prime}+H_{oz}^{ns}(-m_{2})+H_{op}^{ns}(-m_{2})\end{split}  \tag{52}\]  
recalling that all occurrences of \(m_{i}\) are functions of \(\eta\), i.e., \(m_{i}(\eta)\). The integral equations in (52) are of the singular type; for this reason, we resort to the Fredholm factorization method to obtain regularized expressions. The procedure consists of \(\gamma_{1t}\) Cauchy _smile_ contour integration [11] on both sides of each equation and consequent mathematical elaboration. Focusing our attention on the LHS for each term of each equation in (52), using dummy subscripts, we have the following regularized expression:  
\[ \begin{split}\tfrac{1}{2\pi i}\int\limits_{\gamma_{1t}}\tfrac{A(t )H_{+}(t)}{t-\eta}dt&=\tfrac{1}{2\pi i}\int\limits_{\gamma_{1t}} \tfrac{(A(t)-A(\eta))H_{+}(t)}{t-\eta}dt+\tfrac{A(\eta)}{2\pi i}\int\limits_{ \gamma_{1t}}\tfrac{H_{+}(t)}{t-\eta}dt\\ &=\tfrac{1}{2\pi j}\int\limits_{-\infty}^{\infty}\tfrac{(A(t)-A( \eta))H_{+}(t)}{t-\eta}dt+A(\eta)H_{+}(\eta)-A(\eta)H_{+}^{ns}(\eta).\end{split}