## 2 Theoretical Background
### Hollow Waveguide of Arbitrary Cross Section
We assume that the infinite vector of Fourier coefficients \(\xi^{0}=\left\{\xi_{l}^{(0)}\right\}_{l=-\infty}^{\infty}\) to be found lies in \(l_{2}=\left\{\xi_{l}^{(0)}:\ \sum_{l=-\infty}^{\infty}\left|\xi_{l}^{(0)} \right|^{2}<\infty\right\}\). After the replacement of all functions in (7) by their Fourier expansions and performing the integrations, employing the completeness and orthogonality of the exponential functions \(\left\{e^{in\theta}\right\}_{n=-\infty}^{\infty}\) on the interval \([-\pi,\ \pi]\) expressed by the identity  
\[\int_{-\pi}^{\pi}e^{i(l-n)\tau_{0}}\ d\tau_{0}=2\pi\delta_{ln}=2\pi\left\{ \begin{array}{ll}1,&l=n\\ 0,&l\
eq n\end{array} \right. \tag{13}\]  
we arrive at the summation-type equations  
\[\sum_{n=-\infty}^{\infty^{\prime}}\frac{\xi_{n}^{(0)}}{|n|}e^{in\theta_{0}}+ \sum_{n=-\infty}^{\infty}\sum_{p=-\infty}^{\infty}h_{np}^{00}(k)\xi_{-p}^{(0) }e^{in\theta_{0}}=0,\ \theta_{0},\tau_{0}\in[-\pi,\ \pi]. \tag{14}\]  
This is an infinite system of linear algebraic equations (ISLAE) to be solved for the vector \(\xi^{0}\in l_{2}\). Multiplying (14) by \(e^{in\theta_{0}}\) (for a fixed integer \(m\)) and using (13) produces the ISAEs  
\[h_{00}^{00}(k)\xi_{0}^{(0)}+\sum_{p=-\infty}^{\infty^{\prime}}h_{0p}^{00}(k) \xi_{-p}^{(0)}=0,\ m=0 \tag{15}\]  
and  
\[h_{m0}^{00}(k)\xi_{0}^{(0)}+\frac{\xi_{m}^{(0)}}{|m|}+\sum_{p=-\infty}^{ \infty^{\prime}}h_{mp}^{00}(k)\xi_{-p}^{(0)}=0,\ m\
eq 0. \tag{16}\]  
At this stage of the derivation of an equivalent form of the infinite-matrix (summation) equation to be solved, we wish to stress that the analytical semi-inversion is achieved by using only the (inverse) exponential Fourier transform in the process of inversion of the singular part of the initial boundary IE. Note that Equations (15) and (16) are not in the final regularised form (i.e., in the form of the second-kind ISAE) because the coefficients of the terms \(\xi_{m}^{(0)}\) tend to zero as \(|m|\to\infty\). Further transform is effected by introducing new variables \(x_{p},y_{p}\) and rescaled variables \(X_{p},Y_{p}\) defined for positive integers \(p\) by  
\[x_{p}=\frac{1}{2}\Big{(}\xi_{p}^{(0)}+\xi_{-p}^{(0)}\Big{)},y_{p}=\frac{1}{2} \Big{(}\xi_{p}^{(0)}-\xi_{-p}^{(0)}\Big{)};X_{p}=\frac{x_{p}}{\sqrt{p}},Y_{p}= \frac{y_{p}}{\sqrt{p}}. \tag{17}\]  
Note that the introduction of new variables corresponds to re-writing summation Equation (14) in the form with a trigonometric kernel by making the replacement \(e^{in\theta_{0}}=\cos n\theta_{0}+i\sin n\theta_{0}\). Further algebraic transformations (entirely based on readily repeatable simple algebra and omitted for the sake of brevity) produce two coupled ISLAEs of the second kind with respect to the two infinite sequences of unknown Fourier coefficients \(\{X_{m}\}_{m=1}^{\infty}\) and \(\{Y_{m}\}_{m=1}^{\infty}\):  
\[\frac{1}{2}K_{m}^{(+,+)}\xi_{0}^{(0)}+X_{m}+\sum_{p=1}^{\infty}\Big{(}K_{m,p} ^{(+,+)}X_{p}-K_{m,p}^{(+,-)}Y_{p}\Big{)}=0,\ m=1,2,3,\ldots \tag{18}\]  
\[\frac{1}{2}K_{m}^{(-,+)}\xi_{0}^{(0)}+Y_{m}+\sum_{p=1}^{\infty}\Big{(}K_{m,p} ^{(-,+)}X_{p}-K_{m,p}^{(-,-)}Y_{p}\Big{)}=0,\ m=1,2,3,\ldots \tag{19}\]  
The equation corresponding to index \(m=0\) is  
\[\frac{1}{2}\kappa_{0,0}^{(+,+)}\xi_{0}^{(0)}+\sum_{p=1}^{\infty}\Big{(}K_{p}^ {(+,+)}X_{p}-K_{p}^{(+,-)}Y_{p}\Big{)}=0 \tag{20}\]  
Here, we used the notations  
\[