## 4 Asymptotic Estimation of the Field in the Angular Region
### Implementation of a Numerical Example and Validation of Direct Fredholm Factorization
This takes care of the different reference coordinates (see the discussion in Sections 3 and 4 considering region 1 characterized by \(\gamma\) as region \(2^{\prime}\) characterized by \(\pi-\gamma\); see Figures 1 and 2, (45), and (59)):  
\[\tilde{\mathbf{\psi}}_{\mathbf{y}}^{\mathbf{ho}}(\eta,v)=\sum_{i=1}^{2}\mathbf{v}_{i}\cdot \tilde{\mathbf{\psi}}_{\mathbf{y}}(\eta,0)e^{-\lambda_{\eta i}(\gamma)\,v}\mathbf{u}_{i}, \ v>0, \tag{121}\]  
\[\tilde{\mathbf{\psi}}_{\mathbf{Y1}}^{\mathbf{ho}}(\eta,v)=\sum_{i=3}^{4}\mathbf{v}_{\mathbf{\varLambda _{1}}}\cdot\tilde{\mathbf{\psi}}_{\mathbf{Y1}}(\eta,0)e^{-\lambda_{\eta i}\gamma_{1}( \pi-\gamma)\,v}\mathbf{u}_{\mathbf{\varLambda_{1}}},\ v<0. \tag{122}\]  
Starting from the inversion of the contribution of face \(o\) (121), we have  
\[\mathbf{\psi}_{\mathbf{y}}^{\mathbf{ho}}(u,v)=\frac{1}{2\pi}\int_{B_{r}}\tilde{\mathbf{\psi}}_ {\mathbf{y}}^{\mathbf{ho}}(\eta,v)e^{-j\eta u}d\eta. \tag{123}\]  
According to the coordinate mapping in (18), from (16) and (23) we have  
\[-\lambda_{\eta i}(\gamma)\,v-j\eta u=-j\eta\cos\gamma\,v-j\xi_{i}\sin\gamma v- j\eta(x-v\cos\gamma)=-j(\eta x+\xi_{i}y),\,i=1,2 \tag{124}\]  
with \(\xi_{i}=\xi\), \(i=1,2\); thus,  
\[\mathbf{\psi}_{\mathbf{y}}^{\mathbf{ho}}(x,y)=\frac{1}{2\pi}\int_{B_{r}}\sum_{i=1}^{2}\bm {v}_{i}\cdot\tilde{\mathbf{\psi}}_{\mathbf{y}}(\eta,0)\mathbf{u}_{i}e^{-j(\eta x+\xi y)}d\eta, \tag{125}\]  
where \(B_{r}\) is the Bromwich contour (over all singularities). The asymptotic estimation of (125) at the far field is composed of GO terms (captured poles) and GTD diffracted components (due to the saddle point with the application of the SDP method); in global cylindrical coordinates, this is  
\[\mathbf{\psi}_{\mathbf{y}}^{\mathbf{ho}\mathbf{std}}(\rho,\varphi)=\sqrt{\frac{k}{2\pi\rho}}e^ {-j(k\rho-\pi/4)}\sum_{i=1}^{2}\mathbf{v}_{i}\cdot\tilde{\mathbf{\psi}}_{\mathbf{y}}(k \cos\varphi,0)\mathbf{u}_{i}\sin|\varphi|. \tag{126}\]  
In our test problem (region 1 with PEC faces with \(H_{z}\) polarization), this reduces to the third component  
\[\mathbf{\psi}_{\mathbf{y}}^{\mathbf{ho}\mathbf{std}}(\rho,\varphi)[3]=H_{\alpha z}^{\mathbf{std} }(\rho,\varphi)=\sqrt{\frac{k}{2\pi\rho}}e^{-j(k\rho-\pi/4)}\frac{H_{\alpha z} (k\cos\varphi)}{2}\sin|\varphi| \tag{127}\]  
according to the definition of \(\tilde{\mathbf{\psi}}_{\mathbf{y}}(\eta,v)\) in (3) and \(\mathbf{u}_{i},\mathbf{v}_{i}\) in (62)-(63). We obtain the following GTD diffraction coefficient component due to face \(o\):  
\[D_{H\alpha z}^{\mathbf{std}}(\varphi)=\frac{kH_{\alpha z}(k\cos\varphi)\sin| \varphi|}{j2H_{o}}. \tag{128}\]  
Now, we repeat the procedure starting from the inversion of the contribution of face \(a\) (122) using the notation in Figures 1 and 2b:  
Figure 7:0  
According to the coordinate mapping  
Figure 7:1  
from (16) and (23) we have  
Figure 7:2where \(\xi_{i}=\xi\), \(i=3,4\); thus,  
Figure 7:3  
where \(B_{r}\) is a Bromwich contour. The asymptotic estimation of (132) at the far field is composed of GO terms and GTD diffracted components in the following global cylindrical coordinates:  
Figure 7:4  
For our test problem (region 1 with PEC faces with \(H_{z}\) polarization), this reduces to the third component  
Figure 7:5  
according to the definition of \(\bar{\boldsymbol{\psi}}_{\boldsymbol{y}}(\eta,v)\) in (3) and \(\boldsymbol{u}_{i\boldsymbol{\Upsilon_{1}}}=\boldsymbol{u}_{i},\boldsymbol{v} _{i\boldsymbol{\Upsilon_{1}}}=\boldsymbol{v}_{i}\) in (62)-(63). Note that the invariance of \(\boldsymbol{u}_{i\boldsymbol{\Upsilon_{1}}}=\boldsymbol{u}_{i},\boldsymbol{v} _{i\boldsymbol{\Upsilon_{1}}}=\boldsymbol{v}_{i}\) in the rotation of the reference system is allowable only in isotropic regions; otherwise, a more complex procedure is required for their definition in arbitrary linear media (see Section 2).