## 4 Asymptotic Estimation of the Field in the Angular Region  
Having obtained the spectra at the faces of the angular region, we can estimate the asymptotic behavior of the far field inside of the angular region. Going back to the solution of (26) in Section 2 for region 1, according to (28) we have  
\[\boldsymbol{\tilde{\psi}_{y}}(\eta,v)=\sum\limits_{i=1}^{2}C_{i}e^{-\lambda_ {\eta i}(\gamma)\,v}\boldsymbol{u}_{i}+\sum\limits_{i=3}^{4}\boldsymbol{u}_{ i}\boldsymbol{v}_{i}\cdot\int\limits_{v}^{\infty}e^{-\lambda_{\eta i}( \gamma)(v-v^{\prime})}\boldsymbol{\psi_{sa}}(v^{\prime})dv^{\prime},\;v>0. \tag{56}\]  
From the homogeneous portion of the solution in (56), we obtain the definitions of arbitrary coefficients in terms of the field components at \(v=0\) (face \(o\)):  
\[\boldsymbol{v}_{i}\cdot\boldsymbol{\tilde{\psi}_{y}}(\eta,0)=C_{i},\;i=1,2. \tag{57}\]  
The particular integrals in (56) are terms related to face \(a\) via \(\boldsymbol{\psi_{sa}}(v)\). Due to the linearity of the problem, we apply the superposition principle; we can interpret (56) as the result of an equivalent theorem, where \(\boldsymbol{\tilde{\psi}_{y}}(\eta,v)\) is represented through equivalent sources at facesand \(a\). Similarly, the spectral field in region 1 can be considered as a result of the analysis of the rotated region \(2^{\prime}\) from Figure 2b in Section 3, yielding  
\[-\frac{d}{dv}\vec{\psi}_{\mathbf{Y_{1}}}(\eta,v)=\mathbf{M}_{\pi-\gamma}(-j\alpha_{o },-j\eta)\cdot\vec{\psi}_{\mathbf{Y_{1}}}(\eta,v)+\mathbf{\psi}_{\mathbf{so}}(v),\ v<0, \tag{58}\]  
where we note that \(\gamma\to\pi-\gamma\) impacts all terms of the solution, as already reported in Section 3, i.e., \(u_{\mathbf{i}\mathbf{\chi}_{\mathbf{Y_{1}}}},\mathbf{v}_{\mathbf{i}\mathbf{\chi}_{\mathbf{Y _{1}}}},\lambda_{\mathbf{i}\mathbf{\chi}_{\mathbf{1}}},\mu_{\mathbf{i}\mathbf{\chi}_{ \mathbf{1}}}\), and the field components. The solution takes the form  
\[\vec{\psi}_{\mathbf{Y_{1}}}(\eta,v)=\sum_{i=3}^{4}C_{i}e^{-\lambda_{\mathbf{i} \mathbf{\chi}_{\mathbf{1}}}(\pi-\gamma)\,v}u_{\mathbf{i}\mathbf{\chi}_{\mathbf{1}}}- \sum_{i=1}^{2}\mathbf{u}_{\mathbf{i}\mathbf{\chi}_{\mathbf{1}}}v_{\mathbf{i}\mathbf{\chi} _{\mathbf{1}}}\cdot\int\limits_{v}^{\infty}e^{-\lambda_{\mathbf{i}\mathbf{\chi}_{ \mathbf{1}}}(\pi-\gamma)(v-v^{\prime})}\mathbf{\psi}_{\mathbf{so}}(v^{\prime})dv^{ \prime},\ v<0, \tag{59}\]  
where \(v=-x\) in Figure 2b is now different from \(v=X_{1}\) in Figure 2a. From the homogeneous portion of the solution in (59), we obtain the definitions of arbitrary coefficients in terms of the field components at \(v=0\) (face \(a\)):  
\[\mathbf{v}_{\mathbf{i}\mathbf{\chi}_{\mathbf{1}}}\cdot\vec{\psi}_{\mathbf{Y_{1}}}( \eta,0)=C_{i},\ i=3,4. \tag{60}\]  
The particular integrals in (59) are terms related to face \(o\) via \(\mathbf{\psi}_{\mathbf{so}}(v)\). Due to the linearity of the problem, we again apply the superposition principle and can interpret (59) as the result of an equivalent theorem, where \(\vec{\psi}_{\mathbf{Y_{1}}}(\eta,v)\) is represented through equivalent sources at faces \(a\) and \(o\). Using the superposition principle and considering only homogeneous portions of (56) and (59), we can represent the complete field without the particular integrals. Each contribution originating from (56) and (59) is a spectral component that can be inversely Fourier/Laplace transformed into the physical domain \((u,v)\); these contributions represent the fields from equivalent currents distributed in half-planes (face \(o\) and face \(a\), respectively). The application of the asymptotic representation of fields for each component in a unique global system of cylindrical coordinates provides the estimation of the field in terms of the classical GTD for angular region 1, but as a superposition of the GTD for two half-planes (face o and face a), as in Kirchhoff representations.