## 4 Asymptotic Estimation of the Field in the Angular Region
### Classical Solution of the GWHEs of the Problem in Different Complex Planes  
In order to illustrate and validate the new direct Fredholm factorization procedure of Section 3 in the following subsections, this subsection presents the classical WH solution of (68) and (69) obtained in closed form [11] with the help of a specialized mapping, factorization, and decomposition and with the extraction of source terms such as geometrical optics (GO) fields for plane wave illumination. This subsection also clarifies important properties related to different complex planes (including the angular complex plane \(w\)) where the problem and the solutions are represented. The specialized mapping for the closed-form WH solution is  
\[\eta=-k\cos\biggl{(}\frac{\gamma}{\pi}\arccos\biggl{(}-\frac{\eta}{k}\biggr{)} \biggr{)}, \tag{70}\]  
which was introduced for the first time in 2001 and has been extensively used in isotropic wedge scattering problems, as reported in [11; 12]. The mapping transforms plus unknowns in the \(\eta\) plane and minus unknowns in the \(m\) plane (61) into plus and minus unknowns in the \(\bar{\eta}\) plane, respectively, yielding the following classical Wiener-Hopf equations in the new complex plane \(\bar{\eta}\):  
\[H_{o\rho+}(\bar{\eta})=H_{a\rho+}(-\bar{\eta}) \tag{71}\]  
\[\xi H_{oz+}(\bar{\eta})=[\eta\sin(\gamma)+\xi\cos(\gamma)]H_{az+}(-\bar{\eta}) \tag{72}\]  
where \(\xi\) and \(\eta\) become functions of \(\bar{\eta}\) and  
\[m=k\cos\biggl{(}\frac{\gamma}{\pi}\arccos\biggl{(}-\frac{\eta}{k}\biggr{)}+ \gamma\biggr{)}. \tag{73}\]  
From this point, the solution proceeds as it does for CWHEs, with factorization, decomposition, and the application of [PERSON]'s theorem while considering plane wave illumination with \(E_{z}\) and \(H_{z}\) polarization and incident waves. \[E_{z}^{i}(\rho,\varphi)=E_{o}e^{ik\rho\cos(\varphi-\varphi_{o})},\ H_{\rho}^{i }(\rho,\varphi)=-\frac{1}{j\omega\mu\rho}\frac{\partial E_{z}^{i}(\rho, \varphi)}{\partial\varphi}=\frac{k}{\omega\ \mu}\sin(\varphi-\varphi_{o})e^{ik\rho\cos(\varphi-\varphi_{o})}E_{o} \tag{74}\]  
\[H_{z}^{i}(\rho,\varphi)=H_{o}e^{ik\rho\cos(\varphi-\varphi_{o})},\ E_{\rho}^{i }(\rho,\varphi)=\frac{1}{j\omega\varepsilon\rho}\frac{\partial H_{z}^{i}( \rho,\varphi)}{\partial\varphi}=-\frac{k}{\omega\varepsilon}\sin(\varphi- \varphi_{o})e^{ik\rho\cos(\varphi-\varphi_{o})}H_{o} \tag{75}\]  
Due to the PEC boundary conditions, we obtain the following GO source terms, which are tangential to faces \(a\) and \(o\), respectively, of angular region 1. \[H_{x}^{\rm GO}(\rho,0)=-2\frac{E_{o}}{Z_{o}}\sin\varphi_{o}e^{ik\rho\cos\varphi _{o}},\ H_{\rho}^{\rm GO}(\rho,\gamma)=2\frac{E_{o}}{Z_{o}}\sin(\gamma- \varphi_{o})e^{ik\rho\cos(\gamma-\varphi_{o})} \tag{76}\]  
\[H_{z}^{\rm GO}(\rho,0)=2H_{o}e^{ik\rho\cos(\varphi_{o})},\ H_{z}^{\rm GO}( \rho,\gamma)=2H_{o}e^{ik\rho\cos(\varphi-\varphi_{o})} \tag{77}\]  
In the spectral domain, according to the Laplace transforms (33)-(34), (76)-(77) become  
\[H_{o\rho}^{\rm GO}(\eta)=\frac{-2 jE_{o}\sin\varphi_{o}}{Z_{o}(\eta-\eta_{o})},\ H_{a\rho}^{\rm GO}(-m)=\frac{-2 jE_{o}\sin(\gamma-\varphi_{o})}{Z_{o}(m-m_ {o})} \tag{78}\]  
\[H_{oz}^{\rm GO}(\eta)=\frac{2 jH_{o}}{\eta-\eta_{o}},\ H_{az}^{\rm GO}(-m)= \frac{-2 jH_{o}}{m-m_{o}} \tag{79}\]  
with \(\eta_{o}=-k\cos\varphi_{o}\) and \(m_{o}=k\cos(\gamma-\varphi_{o})\). In the \(\bar{\eta}\) plane (70), pole \(\eta_{o}\) is mapped into \(\bar{\eta}_{o}=-k\cos(-\frac{\eta}{\eta}\varphi_{o})\). In the following, we assume \(\varphi_{o}<\gamma/2\) to locate \(\bar{\eta}_{o}\) in the upper half-plane of the complex plane \(\bar{\eta}\), yielding non-standard plus unknowns; generalization is straightforward, yielding \(\bar{\eta}_{o}\) in the \(\bar{\eta}\)-lower half-plane, while \(\gamma/2<\varphi_{o}<\gamma\).