# Wave Diffraction from a Bicone Conjoined with an Open-Ended Conical Cavity
## 4 Mode-Matching Series Equations  
Using the boundary condition (5) and representation (18), we arrive at the relation as  
\[x_{n}^{(2,2)}=x_{n}^{(2,1)}\mathrm{Y}_{n}(\omega), \tag{19}\]  
where  
\[\mathrm{Y}_{n}(\omega)=-\frac{K_{\mu_{n}}(sc_{1})I_{\mu_{n}}(sa_{1})}{K_{\mu_{ n}}(sa_{1})I_{\mu_{n}}(sc_{1})}\Bigg{[}\frac{1+2 sc_{1}K_{\mu_{n}}^{\prime}(sc_{1}) /K_{\mu_{n}}(sc_{1})}{1+2 sc_{1}I_{\mu_{n}}^{\prime}(sc_{1})/I_{\mu_{n}}(sc_{1} )}\Bigg{]}. \tag{20}\]  
Considering the asymptotic properties of the modified Bessel and the Mcdonald functions and representation (20), we find that \(x_{n}^{(2,2)}=O(n^{-2}(a_{1}/c_{1})^{2n})\) if \(n\to\infty\).  
In order to find the unknown expansion coefficients in (18), we use the mode matching of the tangential total fields \(H_{\varphi}^{t}\) and \(E_{\theta}^{t}\) on the spherical surface \(\widehat{S}\): \(\{r=a_{1},\ \gamma_{1}\leq\theta\leq\pi\}\), which covers the circular aperture of a truncated cone. This leads to the functional (series) equations that are defined on \(\theta\in[\gamma_{1},\pi]\) with the associated Legendre functions kernel. Since the series representing the \(E_{\theta}\) field is singular at the edge as \(E_{\theta}(r,\theta)=O(\widehat{\rho}^{-1/2})\), where \(\widehat{\rho}\) is the relative distance to the edge in the local coordinate system, the series are not absolutely convergent on \(\widehat{S}\); their sums depend on the order of summation. Therefore, we depict the mode-matching series in the form of the limit transition as  
\[\lim_{N\to\infty}\sum\limits_{n=1}^{N}P_{z_{n-1/2}}^{1}(-\cos\theta)\bigg{[}x _{n}^{(1)}+\bar{A}_{n}\frac{K_{z_{n}}(sa_{1})}{K_{z_{n}}(sr_{0})}\bigg{]}\]  
\[=\left\{\begin{array}{l}\lim\limits_{P\to\infty}\sum\limits_{p=1}^{P}x_{n}^ {(2)}P_{\mu_{p-1/2}}^{1}(-\cos\theta)\bigg{[}1+\mathrm{Y}_{\mu_{p}}(\omega) \bigg{]},\ \ (r,\theta)\in D_{2},\ p=1,\,\ldots P,\\ \\ -\lim\limits_{K\to\infty}\sum\limits_{k=1}^{K}x_{k}^{(3)}\Psi_{v_{k-1/2}}(\cos \theta),\ \ (r,\theta)\in D_{3},\ k=1,\,\ldots K;\\ \\ \lim\limits_{N\to\infty}\sum\limits_{n=1}^{N}P_{z_{n-1/2}}^{1}(-\cos\theta) \bigg{[}x_{n}^{(1)}\frac{I_{z_{n}}^{\prime}(sa_{1})}{I_{z_{n}}(sa_{1})}+\bar{ A}_{n}\frac{K_{z_{n}}^{\prime}(sa_{1})}{K_{z_{n}}(sr_{0})}\bigg{]}\\ \\ =\left\{\begin{array}{l}\lim\limits_{P\to\infty}\sum\limits_{p=1}^{P}x_{p}^ {(2)}P_{\mu_{p-1/2}}^{1}(-\cos\theta)\bigg{[}\frac{K_{\mu_{p}}^{\prime}(sa_{1} )}{K_{\mu_{p}}(sa_{1})}+\mathrm{Y}_{\mu_{p}}(\omega)\frac{I_{\mu_{p}}^{\prime }(sa_{1})}{I_{\mu_{p}}(sa_{1})}\bigg{]},\ \ (r,\theta)\in D_{2},\,p=1,\,\ldots P,\\ \\ -\lim\limits_{K\to\infty}\sum\limits_{k=1}^{K}x_{k}^{(3)}\Psi_{v_{k-1/2}}(\cos \theta)\frac{K_{v_{k}}^{\prime}(sa_{1})}{K_{v_{k}}(sa_{1})};\ (r,\theta)\in D_{3},\ k=1,\,\ldots K.\end{array}\right.\end{array}\right. \tag{22}\]  
Here, \(N=P+K\), the prime denotes a derivative with respect to the argument. For further convenience, we introduce the new notation as  
\[\bar{A}_{n}=A_{n}I_{z_{n}}(sr_{0})K_{z_{n}}(sr_{0})\ \ \text{and}\ \ x_{n}^{(2)}=x_{n}^{(2 ;1)}.\]  
Equations (21) and (22) are the desirable series equations of our problem. It is worth noting that the rules of the limit transition in (21) and (22) will be determined using the Meixner condition at the aperture's edge.  
It is well known that there are different ways to reduce the mode-matching equations to the infinite system of linear algebraic equations (ISLAE) using the orthogonality properties of the eigenfunctions. As a result, we arrive at different kinds of the ISLAE. In order to carry out the analytical regularization, we choose the ISLAE acceptable for the extraction of the singular part of their matrix operators.