# Wave Diffraction from a Bicone Conjoined with an Open-Ended Conical Cavity
## 5 The ISLAE of the First Kind
### Equations with the Unknowns \(\{x_{n}^{(1)}\}_{n=1}^{\infty}\)
Equations with the Unknowns \(\{x_{n}^{(2)}\}_{n=1}^{\infty}\) and \(\{x_{n}^{(3)}\}_{n=1}^{\infty}\)  
Let us consider the representation of the associated Legendre functions of the form  
\[\left. \begin{array}{l}\Psi_{x_{k}-1/2}(\cos\theta),\ \gamma_{1}\leq\theta< \gamma_{2}\\ 0,\qquad\qquad\gamma_{2}<\theta\leq\pi\\ 0,\qquad\qquad\gamma_{1}\leq\theta<\gamma_{2}\\ \left.\begin{array}{l}P_{\mu_{p}-1/2}^{1}(-\cos\theta),\ \gamma_{2}<\theta\leq\pi\\ \end{array}\right\}=\tilde{q}^{-}(\mu_{p},\gamma_{2})\lim_{N\to\infty}\sum_{ n=1}^{N}\frac{\bar{\alpha}(z_{n},\gamma_{1},\gamma_{2})}{\mu_{p}^{2}-z_{n}^{2}}P_{z_{ n}-1/2}^{1}(-\cos\theta),\ \gamma_{1}\leq\theta\leq\pi,\\ \mbox{where }k,\ p=1,\,2,\,3,\,\ldots;\\ \tilde{q}^{+}(v_{k},\gamma_{2})=\pi\sin\gamma_{2}\Psi_{v_{k}-1/2}(\cos\gamma _{2}),\\ \tilde{q}^{-}(\mu_{p},\gamma_{2})=\pi\sin\gamma_{2}P_{\mu_{p}-1/2}^{1}(-\cos \gamma_{2}),\\ \tilde{\alpha}(z_{n},\gamma_{1},\gamma_{2})=\frac{z_{n}P_{z_{n}-1/2}(\cos \gamma_{1})P_{z_{n}-1/2}(-\cos\gamma_{2})}{\cos(\pi z_{n})\tilde{\theta}_{1}[ P_{t-1/2}(-\cos\gamma_{1})]_{\,t=z_{n}}}.\end{array} \right. \tag{41}\]  
Here, we use the orthogonality properties of the associated Legendre functions \(P_{z_{n}-1/2}^{1}(-\cos\theta)\) over the interval \(\gamma_{1}\leq\theta\leq\pi\) to obtain these expressions [46]. It is worth noting that series (39) and (40) are formal because the asymptotic behaviour of the moduli of their terms decays as \(n^{-1}\) if \(n\ \to\infty\). Therefore, we substitute series (39) and (40) into the right-hand side of Equations (21) and (22), taking into account the finite numbers of their terms. Then, let us form the homogeneous series equations concerning kernel functions \(P_{z_{n}-1/2}^{1}(-\cos\theta)\) and equate the coefficients to zero. Therefore, we arrive at the finite linear algebraic system as  
\[\sum_{k=1}^{K}\frac{y_{k}^{(3)}}{v_{k}^{2}-z_{n}^{2}}+\sum_{p=1}^{P}\frac{y_{p }^{(2)}}{\mu_{p}^{2}-z_{n}^{2}}\Big{[}1+\mbox{Y}_{\mu_{p}}(\omega)\Big{]}= \frac{1}{\bar{\alpha}(z_{n},\gamma_{1},\gamma_{2})}\bigg{[}x_{n}^{(1)}+\bar{A }_{n}\frac{K_{z_{n}}(sa_{1})}{K_{z_{n}}(sr_{0})}\bigg{]}, \tag{42}\]  
\[\sum_{k=1}^{K}\frac{y_{k}^{(3)}}{v_{k}^{2}-z_{n}^{2}}\frac{K_{v_{k}}^{\prime}( sa_{1})}{K_{v_{k}}(sa_{1})}+\sum_{p=1}^{P}\frac{y_{p}^{(2)}}{\mu_{p}^{2}-z_{n}^{2}} \bigg{[}\frac{K_{\mu_{p}}^{\prime}(sa_{1})}{K_{\mu_{p}}(sa_{1})}+\mbox{Y}_{\mu _{p}}(\omega)\frac{I_{\mu_{p}}^{\prime}(sa_{1})}{I_{\mu_{p}}(sa_{1})}\bigg{]} \tag{43}\]  
\[=\frac{1}{\bar{\alpha}(z_{n},\gamma_{1},\gamma_{2})}\bigg{[}x_{n}^{(1)}\frac{I _{z_{n}}^{\prime}(sa_{1})}{I_{z_{n}}(sa_{1})}+\bar{A}_{n}\frac{K_{z_{n}}^{ \prime}(sa_{1})}{K_{z_{n}}(sr_{0})}\bigg{]}.\]  
Here, \(n=1,\,2,3,\ldots,N;N=K+P\);  
\[y_{k}^{(3)}=-x_{k}^{(3)}\tilde{q}^{+}(v_{k},\gamma_{2}), \tag{44}\]  
\[y_{p}^{(2)}=x_{p}^{(2)}\tilde{q}^{-}(\mu_{p},\gamma_{2}). \tag{45}\]  
Taking into account that \(\tilde{q}^{+}(v_{k},\gamma_{2})=O(1)\) and \(\tilde{q}^{-}(\mu_{p},\gamma_{2})=O(p^{1/2})\) if \(k\), \(p\to\infty\), we find that \(y_{n}^{(3)},y_{n}^{(2)}=O(n^{-3/2})\). Let us exclude the unknowns \(x_{n}^{(1)}\) from the right-hand part of Equations (42) and (43). Then, we pass to the limit \(N,P,K\to\infty\ \ (N=P+K)\) and arrange the unknowns according to the sequence (34). This leads to the ISLAE  
\[(\mathbf{C_{11}}+\mathbf{D_{11}})\mathbf{Y}=\mathbf{\Phi}. \tag{46}\]  
Here, \(\mathbf{Y}:\{y_{q}\}_{q=1}^{\infty}\) is the unknown vector,\[y_{q}=\left\{ \begin{array}{ll}y_{k}^{(3)},&\xi_{q}\in\{\
u_{k}\}_{k=1}^{\infty}, \\ y_{p}^{(2)},&\xi_{q}\in\{\mu_{p}\}_{p=1}^{\infty},\end{array} \right. \tag{47}\]  
\(\mathbf{C_{11}:}\{c_{nlq}^{(1,1)}\}_{n,q=1}^{\infty}\), \(\mathbf{D_{11}:}\{d_{nlq}^{(1,1)}\}_{n,q=1}^{\infty}\) are infinite matrices with the elements  
\[c_{nlq}^{(1,1)}=\frac{sa_{1}W[K_{\xi_{q}}l_{z_{n}}]_{sa_{1}}}{(\xi_{q}^{2}-z_{ n}^{2})K_{\xi_{q}}(sa_{1})l_{z_{n}}(sa_{1})}; \tag{48}\]  
\[d_{nlq}^{(1,1)}=\left\{