## 4 Asymptotic Estimation of the Field in the Angular Region
### Implementation of a Numerical Example and Validation of Direct Fredholm Factorization  
Let us consider region 1 in Figure 1 with an aperture angle \(\pi/2<\gamma<\pi\), which is filled by a homogeneous isotropic medium with a propagation constant \(k\) (\(k=k_{r}-jk_{i}\), \(k_{i}<<k_{r}\)) and terminated by PEC boundary conditions. The angular region is illuminated by an \(H_{z}\) polarized plane wave with an incoming direction \(\varphi_{o}\) (\(0<\varphi_{o}<\pi-\gamma\)) and intensity \(H_{o}\). The spectral solution \((H_{oz}(\eta),H_{az}(\eta))\) can be provided by the system of FIEs in (108)-(109). Due to the convergence properties of the kernel [41], simple sample-and-hold approximation is enforced with the truncation of integration intervals at \(\pm A\) and integration step \(h\) such that \(A/h\in\mathbb{N}\). We tested our novel direct FIE solution against the classical exact closed-form solution provided in Section 5.1 in the \(\eta\) and \(w\) planes, respectively ((88) and (89)). Furthermore, we compared the asymptotic results in terms of GTD coefficients. We examine the case where \(\gamma=0.7\pi\), \(k=1-j0.1\), \(H_{o}=1A/m\), and \(\varphi_{o}=0.1\pi\) in detail. Because we have \(0<\varphi_{o}<\pi-\gamma\), the GO field is constituted by the incident waves, waves reflected by face \(a\), and doubly-reflected waves (from face \(o\) and then from face \(a\)); only the plus spectral unknown along face \(o\), i.e., \(H_{oz}(\eta)\), is nonstandard in the WH formulation (91), as reported in the example in the previous subsection. To enhance the convergence of the approximate FIE solution provided by (108)-(109), we warp the integration line constituted by the real axis into a straight line located in the first and third quadrants of the complex plane at an angle \(\theta\) with respect to the real axis (the singularities of the kernel and the sources are located in the second and fourth quadrants; see Sections 5.3 and 5.4 of [12]):  
\[\alpha_{t}(t)=t\exp^{\beta},\,t\in\mathbb{R},\,0<\theta<\pi/2. \tag{117}\]  
According to the physical parameters of the test problem, we have \(\eta_{ra}=0.309017-0.0309017j\), \(\eta_{raro}=0.809017-0.0809017j\), with both located in the lower half-plane, as considered in (116). The discretization of (108)-(109) by the sample-and-hold technique with \(A,h\) yields a linear system of equations with dimensions of \(2(2(A/h)+1)\):  
\[\left| \begin{array}{cc}\underline{d}_{G}+\underline{K}_{G}&- \underline{M}_{G}\\ -\underline{M}&\underline{d}_{G}+\underline{K}_{G}\end{array} \right|\,\left| \begin{array}{cc}\underline{H}_{\alpha z}\\ \underline{H}_{\alpha z}\end{array} \right|=\left| \begin{array}{c}\underline {s}_{1}\\ \underline{s}_{2}\end{array} \right| \tag{118}\]  
where the diagonal matrix \(\underline{d}_{G}\), the matrix \(\underline{K}_{G}\), and the matrix \(\underline{M}_{G}\) contain samples of \(G(\eta)\), \(\frac{G(t)-G(\eta)}{t-\eta}\), and \(M_{e}(\eta,\eta^{\prime})\), respectively, while the vectors \(\underline{H}_{\alpha z}\), \(\underline{H}_{\alpha z}\) and \(\underline{s}_{1}\), \(\underline{s}_{2}\) contain samples of \(H_{\alpha z}(\eta)\), \(H_{\alpha z}(\eta)\) and \(s_{1}(\eta)\), \(s_{2}(\eta)\), respectively. Note that \(\underline{M}_{G}\) is a coupling matrix that is much weaker than the remaining terms.