## 4 Asymptotic Estimation of the Field in the Angular Region
### Implementation of a Numerical Example and Validation of Direct Fredholm Factorization
The sampled solutions allow for a representation of \(H_{\alpha z}(\eta)\), \(H_{\alpha z}(\eta)\) to be built by substituting them into the integral part of (108)-(109):  
\[H_{\alpha z,\alpha z}(\eta)=-\frac{h}{2\pi j}\sum_{-A/h}^{A/h}\frac{[G^{-1}( \eta)G(\alpha_{t}(hi))-1]H_{\alpha z,\alpha z}(\alpha_{t}(hi))}{\alpha_{t}(hi )-\eta}+\frac{hG^{-1}(\eta)}{(2\pi j)^{2}}\sum_{-A/h}^{A/h}M_{e}(\eta,\alpha_ {t}(hi))H_{\alpha z,\alpha z}(\alpha_{t}(hi))+G^{-1}(\eta)s_{1,2}(\eta). \tag{119}\]  
These approximate expressions of \(H_{\alpha z}(\eta)\), \(H_{\alpha z}(\eta)\) are valid for analytic continuation in the proper sheet of the \(\eta\) plane, which is useful for correctly estimating fields in the physical domain through the asymptotics of half-planes, as discussed in Section 4. This property limits the requirement to know the spectra only in the proper sheet, as acquired in the procedure, which is a novelty and represents progress with respect to classical Fredholm factorization combined with spectral mapping in GWHE wedge problems. To highlight the performance of this method, we compare the spectra along the real axis of the \(\eta\) plane and a segment of the \(\eta\) plane that is useful for asymptotics according to the steepest descent path (SDP) method, which in an isotropic medium corresponds to \(\eta=-k\cos w\) with \(-\pi<w<0\), i.e., the segment that connects \(-k\) with \(k\). To study the convergence of the method, we select physical parameters of region 1 with an aperture angle of \(\gamma=0.7\tau\) and plane wave illumination at \(H_{z}\) polarization with \(H_{o}=1A/m\), \(\varphi_{o}=0.1\tau\), and \(k=1-j0.1\). We select the quadrature parameters \(5\leq A\leq 40,\,0.2\leq h\leq 0.25,\,\theta=0.1\) such that \(A/h\in\mathbb{N}\). The numerical results are provided in Figure 6 along the segment for asymptotic estimation. In the figure, we note that along the segment we have a degradation of the spectral solution near \(w=-\pi,0\), which corresponds to the branch points \(\eta=k,-k\) of \(\xi\) (where \(\xi\) defines the proper and improper sheet of the \(\eta\) plane [11]). We recall that the solution of the FTEs was obtained through the simple sample-and-hold quadrature and estimation of \(M_{e}(\eta,\eta^{\prime})\), which saturates the precision, particularly near \(\eta=k,-k\) (the branch point \(\eta=-k\) is a local offending singularity for the plus spectra that should not appear, while \(\eta=k\) is related to physical structural properties of the problem). An improvement would be obtained with a specialized quadrature (and method of moments) capable of taking into account non-algebraic behavior such as branch points (see [42]); however, the scope of the present method involves obtaining a very simple, fast, and convergent solution that cannot incorporate sophisticated quadratures. Furthermore, we observe that the lack of precision near \(\eta=k,-k\) is mitigated while computing asymptotics because plus spectral unknowns are multiplied by \(\sin w\), providing locally-smoothing errors. However, while the offending \(\eta=-k\) is a very local perturbation, the physical \(\eta=k\) is more present, as it is physical. To recover the quality of the solution near \(w=-\pi,0\) (\(\eta=k,-k\)), we resort to spectral considerations based on the properties of the original GWHE formulation in (91). Equation (91) can be applied to the approximate solutions obtained from the FIEs to obtain a new improved representation of the plus spectra. This application allows us to obtain an approximate spectrum near \(w=-\pi,0\) (\(\eta=k,-k\)) that originates from another portion of the \(\eta\) plane according to \(m(\eta)\). This procedure is particularly effective and valid because \(m(\eta)\) for \(\eta\) in the proper sheet lies in just a portion of the proper sheet of the \(\eta\) plane. This property is particularly evident (in this simplified isotropic problem) by rewriting (91) in the \(w\) plane:  
\[