# Wave Diffraction from a Bicone Conjoined with an Open-Ended Conical Cavity
## 7 Set of the Regularizing Operators  
In this section, we generalize our theory and introduce a set of the regularizing operators that can simplify the regularization procedure. Let there be a given set of even meromorphic functions  
\[\Re:\{\widehat{M}\ (\
u)=\widehat{Q}\ (\
u)/\ \widehat{P}\ (\
u)\},\]  
where \(\widehat{Q}\ (\
u)\) and \(\widehat{P}\ (\
u)\) are integers, even functions of the exponential type that are not equal to zero at \(\
u\ =0\). We will denote the growing sequences of their positive simple non-coinciding roots \(\{\widetilde{z}_{n}\}_{n=1}^{\infty}\) and \(\{\widetilde{\xi}_{q}\}_{q=1}^{\infty}\). Let \(\widetilde{z}_{n}\) and \(\widetilde{\xi}_{q}\), as functions of indices, satisfy the following asymptotic estimations:  
\[|\,z_{n}-\ \widetilde{z}_{n}\ |\ =\ O(1/n),|\,\xi_{q}-\ \widehat{\xi}_{q}\ |\ =\ O(1/q)\,\ n,\ q\ \to\infty, \tag{80}\]  
where \(z_{n}\) and \(\xi_{q}\) are simple zeros and poles of function (60). Taking these into account, we form a set of couples of matrix operators \(\Im_{1}\):\(\widehat{\mathbf{A_{1}}}\widehat{\mathbf{A_{1}}}\widehat{\mathbf{A_{1}}}\) and \(\Im_{2}\):\(\widehat{\mathbf{C_{1}}}\widehat{\mathbf{C_{1}}}\widehat{\mathbf{C_{1}}}\) whose elements are written similarly to those represented by the couples (56), (57) and (64), (65), respectively using the functions \(\widehat{M}\) (\(\
u\)) from \(\Re\). Further, using the matrix operators (56) and (64), we formally introduce two sets of equations  
\[\mathbf{A_{1}X}=\mathbf{F_{1}}\] (81a) and \[\mathbf{C_{1}Y}=\mathbf{\Phi_{1}}, \tag{81b}\]  
where \(\mathbf{F_{1}}:\{f_{q}^{(1)}\}_{q=1}^{\infty}\), \(\mathbf{\Phi_{1}}:\{\varphi_{q}^{(1)}\}_{q=1}^{\infty}\) are known vectors. Then, let us formulate the following statement:  
**Statement. ** An arbitrary couple of operators from \(\Im_{1}\) and \(\Im_{2}\) are the regularizing ones for the ISAEs (35) and (46), respectively. Proof of Statement. : Let us apply the regularization procedure (73) to Equation (81) using the operators from \(\Im_{1(2)}\). This leads to the ISAE of the second kind, which we write as:  
\[x_{n}+\sum_{p=1}^{\infty}b_{np}^{(1)}x_{p} = \eta_{n}^{(1)}\] (82a) and \[y_{n}+\sum_{k=1}^{\infty}b_{nk}^{(2)}y_{k} = \eta_{n}^{(2)}. \tag{82b}\]  
Here,  
\[b_{np}^{(1)}=\sum_{k=1}^{\infty}\widetilde{\tau}_{nk}^{(1)}\ \frac{( \widetilde{\xi}_{k}-\xi_{k})+(z_{p}-\ \widetilde{z}_{p})}{(\widetilde{\xi}_{k}-\ \widetilde{z}_{p})(\xi_{k}-z_{p})},\] \[b_{nk}^{(2)}=\sum_{p=1}^{\infty}\widetilde{\tau}_{np}^{(2)}\ \frac{( \widetilde{\xi}_{k}-\xi_{k})+(z_{p}-\ \widetilde{z}_{p})}{(\widetilde{\xi}_{k}-\ \widetilde{z}_{p})(\xi_{k}-z_{p})},\] \[\eta_{n}^{(1)}=\sum_{k=1}^{\infty}\widetilde{\tau}_{nk}^{(1)}\ f_{k}^{(1)}<\infty,\ \ \eta_{n}^{(2)}=\sum_{p=1}^{\infty}\widetilde{\tau}_{np}^{(2)}\ \varphi_{p}^{(2)}<\infty,\]  
\(\widetilde{\tau}_{np}^{(1)}\) and \(\widetilde{\tau}_{np}^{(2)}\) are matrix elements of the operators \(\widetilde{\mathbf{A_{1}}}\) and \(\widetilde{\mathbf{C_{1}}}\), respectively. Accounting to relations (63), (68), and (80), we establish by direct verification that \(b_{np}^{(1)}=O(n^{-1/2}p^{-2})\) and \(b_{nk}^{(2)}=O(n^{-3/2}k^{-2})\) if \(n,\ p,\ k\to\infty\). Therefore, the solution of Equations (82a) and (82b) exist in \(b(\sigma_{1}=1/2\,)\) and in \(b(\sigma_{1}=3/2\,)\), respectively, and the couples operators \(\widetilde{\mathbf{A_{1}}}\), \(\widetilde{\mathbf{A_{1}}}^{-1}\) and \(\widetilde{\mathbf{C_{1}}}\), \(\widetilde{\mathbf{C_{1}}}^{-1}\) from \(\Im_{1}\) and \(\Im_{2}\) are regularizing ones for Equations (81a) and (81b). It follows directly from this that an arbitrary couple of operators from \(\Im_{1}\) and \(\Im_{2}\) are also regularizers for the ISAEs (35) and (46).