## 2 Spectral Functional Equations in an Angular Region Filled by Arbitrary Linear Media
where \(\mathbf{\psi_{y}}(u,v)\doteq\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}\mathbf{\psi_{y}}( \eta,v)e^{-j\eta u}d\eta\) and  
\[\mathbf{M_{\gamma}}(-j\alpha_{o},-j\eta)=\mathbf{M_{\gamma 0}}-j\eta\mathbf{M_{\gamma 1}}- \eta^{2}\mathbf{M_{\gamma 2}}, \tag{21}\] \[\mathbf{M_{\gamma 0}}=\mathbf{M_{y0}}\sin\gamma,\quad\mathbf{M_{\gamma 1}}=\mathbf{M_{y1}} \sin\gamma-\mathbf{I_{t}}\cos\gamma,\quad\mathbf{M_{\gamma 2}}=\mathbf{M_{y2}}\sin\gamma. \tag{22}\]  
Based on the link between \(\mathbf{M_{\gamma}}\) and \(\mathbf{M_{y}}\), we have \(\mathbf{M_{\gamma}}\) with the same eigenvectors \(\mathbf{u_{i}}\) of \(\mathbf{M_{y}}\), and the following relationship between the eigenvalues \(\lambda_{\gamma i}(\gamma)\) and \(\lambda_{i}\) can be obtained:  
\[\lambda_{\gamma i}(\gamma)=j\eta\cos\gamma+\lambda_{i}\sin\gamma\,\ i=1\cdots 4 \tag{23}\]  
resulting in the following \"oblique\" \(v\)-longitudinal propagation constants:  
\[m_{i}(\gamma) =-j\lambda_{\gamma i}(\gamma)=+\eta\cos\gamma+\xi_{i}\sin\gamma\,\ i=1,2 \tag{24}\] \[m_{i}(\gamma) =+j\lambda_{\gamma i}(\gamma)=-\eta\cos\gamma+\xi_{i}\sin\gamma\, \ i=3,4 \tag{25}\]  
which is in agreement with the relationship between \(\lambda_{i}\) and \(\xi_{i}\) as well as with the correlated progressive and regressive propagating interpretation along the longitudinal direction \(y\) and along the oblique \"longitudinal\" direction \(v\) (progressive/regressive \(e^{\mp jm_{i}\varphi}\)). We note that the quantities \(\mathbf{M_{\gamma}}(-j\alpha_{o},-j\eta)\), \(\lambda_{\gamma i}(\gamma)\), and \(m_{i}(\gamma)\) depend on the geometrical parameter \(\gamma\) and on the spectral variable \(\eta\). With reference to region 1 in Figure 1a, we obtain the functional equations with a circuital interpretation as a mathematical manipulation of the solution of the differential Equation (19) using the Laplace domain \(\mathbf{\tilde{\psi_{y}}}(\eta,v)\doteq\int\limits_{0}^{\infty}e^{j\eta\cdot u} \mathbf{\psi_{y}}(u,v)du\). \[-\frac{d}{dv}\mathbf{\tilde{\psi_{y}}}(\eta,v)=\mathbf{M_{\gamma}}(-j\alpha_{o},-j\eta )\cdot\mathbf{\tilde{\psi_{y}}}(\eta,v)+\mathbf{\psi_{sa}}(v),\ v>0 \tag{26}\]  
\[-\frac{\partial}{\partial v}\psi_{\mathbf{y}}(u,v)=M_{\gamma}(-j\alpha_{o},\frac {\partial}{\partial u})\cdot\mathbf{\psi}_{\mathbf{y}}(u,v). \tag{19}\]0  
The benefit of using the Laplace transform is correlated with the incorporation of boundary conditions through initial conditions with the term \(\mathbf{\psi_{sa}}(v)\). In (26)-(27), the condition \(u=0_{+},v>0\) imposes boundary conditions on the fields along face \(a\) in Figure 1a. The solution is obtained by using the characteristic Green function procedure [34] in terms of homogeneous and particular solutions, yielding the representation shown below. \[-\frac{\partial}{\partial v}\psi_{\mathbf{y}}(u,v)=M_{\gamma}(-j\alpha_{o},\frac {\partial}{\partial u})\cdot\mathbf{\psi}_{\mathbf{y}}(u,v). \tag{19}\]1  
Now, considering the asymptotic behavior in (28) of exponential functions in \(v\) for \(v\to+\infty\), we need to have \(C_{3}=C_{4}=0\); at the same time, the first couple of integrals are null, as \(Re[\lambda_{1,2}]>0\) and \(Re[\lambda_{3,4}]<0\) are related to progressive and regressive waves, respectively.