## 4 Asymptotic Estimation of the Field in the Angular Region
### Implementation of a Numerical Example and Validation of Direct Fredholm Factorization
Finally, we obtain the following GTD diffraction coefficient component due to face \(a\):  
Figure 7:6  
The complete GTD coefficient is just the sum for the superposition of (128) and (135):  
Figure 7:7  
Figure 8 shows the GTD diffraction coefficient for the test problem under consideration, that is, \(\gamma=0.7\pi\) and plane wave illumination with \(H_{z}\) polarization with \(H_{o}=1A/m\), \(\varphi_{o}=0.1\pi\), and \(k=1-j0.1\). The figure reports the exact GTD coefficient in terms of the absolute value and phase together with those obtained with the FIE-approximated estimation of the spectra without and with the application of an iteration while selecting \(A=20,h=0.05,\theta=0.1\). Figure 9 shows the corresponding relative error of the GTD diffraction coefficient on a log\({}_{10}\) scale. We note, as expected, that the solution with the iteration is correct, while the solution without the iteration is lacking in its estimation near the faces of the angular regions, i.e., face \(o\) for \(\varphi=0\) and face \(a\) for \(\varphi=\gamma\), as they are related to the spectra of \(H_{oz}(\eta)\) near \(\eta=k\) (128) and \(H_{az}(\eta)\) near \(\eta=k\) (135), respectively (\(\eta=k\) corresponds to \(w=-\pi\), and is a physical branch point). In addition, note that the spectra of \(H_{oz}(\eta),H_{az}(\eta)\) near \(\eta=-k\) (\(w=0\)) are not used for the GTD computation; thus, the lack of possible precision in the offending branch point does not impact the quality of the solution. Moreover, the change in the slope and level of the relative error in Figure 9 is obtained using the reported algorithm to improve the quality of the approximate solution provided by the direct application of Fredholm factorization. In fact, FIE-iteration involves computation of the GTD diffraction coefficient in (136) via (128) and (135), where the spectra \(H_{oz}(-k\cos w)\) and \(H_{az}(-k\cos w)\) are obtained by enforcing (120) on the approximate spectra obtained through the direct application of Fredholm factorization. This procedure mixes the spectral resolution properties of the two faces, thereby improving the quality of the spectra and recovering the degradation of the spectral resolution near \(w=\pi,0\), i.e., \(\eta=k,-k\). Finally, we comment that the direct implementation of FIEs in the \(w\) plane yields high-precision results in isotropic angular region problems [12], exceeding the precision of the current procedure in terms of spectra; however, we recall that the scope of the present work involves presenting an effective procedure for computing diffraction that is implementable in problems where the \(w\) plane cannot be defined, such as in arbitrarily linear media. An Example of the Application of the Functional Equations in Complex Media: Scattering from a PEC Half-Plane in a Gyrotropic Medium  
The scattering of an electromagnetic plane wave at normal incidence by a perfectly conducting semi-infinite screen embedded in a homogeneous gyrotropic medium (such as plasma) is presented in this section with the goal of validating the proposed method, the functional equations, and the WH equations in nonisotropic media. As our formulation is in terms of field components, we selected [15; 16; 20; 21] as studies for comparison; we did not consider other literature where definitions in terms of potentials were employed. In particular, we selected [15], where the distinguished axis of the electric gyrotropic medium is parallel to the edge of the half-plane, i.e., as in plasma, with a uniform magnetic field  
Figure 7:8 GTD diffraction coefficient (absolute value and phase) for the test problem under consideration: \(\gamma=0.7\pi\) and plane wave illumination with \(H_{z}\) polarization with \(H_{o}=1A/m\), \(\varphi_{o}=0.1\pi\), and \(k=1-j0.1\). The figure reports the exact GTD coefficient together with those obtained with the FIE-approximated estimation of the spectra without and with the application of an iteration while selecting \(A=20,h=0.05,\theta=0.1\).