###### Abstract
For this class, the apparatus of the classical method of separation of variables for the solution of the Helmholtz equation, together with the use of the addition theorems (see, for example, [3; 4; 5; 6]), is a natural, although not unique, way of analyzing the problem. In [7], the generalized spectral-domain (GSD) technique is used to analyze rectangular waveguides with metal inserts of rectangular and circular cross section. In [8], the authors described a method which appears to be suitable to estimate the frequency parameters of any arbitrarily shaped waveguide, provided that the transformation function needed to map the section into a circular region can be obtained. The application of the finite-difference method in the time domain for obtaining the cut-off frequencies of TE and TM modes in waveguides of arbitrary cross section is described in [9]. Polygonal metal waveguides are analyzed analytically and numerically in [10], where the authors used finite difference time domain models of microwave waveguide junctions. Conformal transformation is one of the elegant mathematical tools widely applied in the analysis of waveguides of circular [11; 12; 13] and rectangular cross section [14; 15]. It is pertinent to make the two following observations. There is no formal distinction between the calculated cut-off wave numbers for propagating modes in waveguides and the eigenvalues calculated for standing waves in two-dimensional cavity resonators. Also, despite the differing physical nature of sound and electromagnetic waves, the problems in acoustics for waveguides with sound-soft walls are mathematically equivalent to those concerning transverse magnetic (TM) polarized waves in waveguides of the same cross section with ideally conducting walls. From this point of view, the results obtained in acoustics (see, for example [16; 17; 18; 19; 20]) are directly applicable to electromagnetic studies in waveguides of the same shape, a fact that seems somewhat under-exploited by both acoustical and radio-engineering communities. Looking at the refereed papers in the References, one observes that, in most of them, the calculation of cut-off wave numbers has been performed with accuracy not exceeding 3 to 4 significant decimal digits. More importantly, there is no proof of the correctness of the calculated cut-off wavenumbers; the only validation is via comparison with previously published results obtained by alternative methods. This seems to be largely true of the scientific literature in the field: the paper [21] (employing a Method of Moments approach) is perhaps typical. From this perspective, the paper [5] is distinctive because of its scrupulous approach to the estimation of computational accuracy. The three requirements outlined above are important in practical applications. An example occurs in the publication [22] in which the design of a class of ring-resonator-loaded waveguide notch filters depends upon an accurate determination of the cut-off wavenumbers in the waveguide. Recognizing the practical significance of the calculations, appearing in numerous publications, of cut-off wavenumbers for waveguides with variously shaped outer and inner conductors, there is a need to develop a universal and accurate instrument for providing the ultimate estimate of the correctness of such calculations. This means obtaining a benchmark solution which may be used as an external and independent estimate for other methods. Of more consequence is the goal of calculating the cut-off frequencies of complex waveguide structures with guaranteed accuracy; as a by-product, estimates of the accuracy of calculation by other methods immediately become available. In this paper, we developed an approach to this category of problems. The new solution employs the basic strategy of the rigorous Method of Analytical Regularization (MAR). It was originally developed for rigorous solution of acoustic and electromagnetic wave diffraction problems for open structures of canonical shape with sharp edges (see, for example [23]).