## 6 Conclusions  
A rigorously correct mathematical approach has been developed and applied to the important radio-engineering problem of high-precision calculation of the cut-off wavenumbers for waveguides of arbitrary cross sections with inner conductors. The initial BVP may be formulated as a boundary IE with a (complex) spectral parameter \(\mathbf{k}\), and the problem may be considered as finding the characteristic numbers for an integral operator-valued function (OVF) or its infinite-matrix (summation) OVF. The Method of Analytical Regularization (_MAR_) provides an effective tool in this process: its direct application to the ill-conditioned integral equations derived from a single-layer density representation transforms the problem to the determination of non-trivial solutions of a homogeneous well-conditioned system of coupled infinite matrix equations. \begin{table}
\begin{tabular}{c c} \hline \hline
**Truncation Number \(N\)** & **Computed Value \(\gamma^{(N)}\) of \(\gamma\)** \\ \hline
128 & 91.02652053403598 \(-\) i \(\cdot\) 0.000032228513004 \\ \hline
256 & 91.022057188386270 \(-\) i \(\cdot\) 0.001314350150315 \\ \hline
512 & 91.020887459853000 \(-\) i \(\cdot\) 0.000012035120209 \\ \hline
1024 & 91.018051214398398 \(-\) i \(\cdot\) 0.000000181342561 \\ \hline
2048 & 91.018431876293164 \(-\) i \(\cdot\) 0.00000002809210 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Convergence of the approximations to the wavenumber \(\gamma\) of the mode in the vicinity of \(ka=91.015\) as a function of truncation number. Figure 13: Spectral dependence of the condition number _cond_\(A_{N}(ka)\) computed with truncation number \(N=512\) for the circular waveguide with an eccentrically embedded circular rod (\(b/a=d/a=0.25\)). The second kind nature of the system guarantees that the solution computed for the system truncated to a system of finite order converges to the exact solution of the infinite system as the truncation number increases indefinitely. In practice, the solutions of the finite (truncated) systems exhibit fast convergence. Calculation of the matrix elements with the required accuracy is realized by use of the Fast Fourier Transform (FFT), making the filling of the matrix an extremely fast and convenient procedure. A central role in the investigation of the spectrum of cut-off wavenumbers is played by the spectral characteristics of the condition number of the truncated matrices. These computations allow us to find quite accurate initial approximations to the cut-off wavenumbers, which can be subsequently refined to highly accurate values. This is repeatedly exemplified in the results obtained in Sections 3 and 4. Furthermore, the high-resolution power of this spectral characteristic approach ensures that no modes are missed, even though their cut-off wavenumbers are separated by extremely tiny amounts on the wavenumber axis. Section 3 is devoted to the comparison with, and cross validation of, previously published results obtained by different methods: the hollow elliptical waveguide, the circular waveguide with an inner conductor eccentrically placed, and a circular-rectangular waveguide with one inner circular conductor. In the latter case, the superior accuracy of our approach highlighted the limited accuracy of the previously published results. Because of the second-kind nature of the system of equations obtained by the MAR, we wish to claim that our computed solutions, the accuracy of which are the result of a process with guaranteed convergence, should be regarded as benchmark solutions for the results obtained in this Section. As an application, Section 4 presented a careful analysis of the cut-off wavenumbers in rectangular waveguides with an inner conductor where an infinitesimally thin strip is replaced by a strip of non-zero thickness. Two classes of thickened strips were considered: those of elliptical cross section and a rectangular thickened strip (with rounded corners).