## 2 Theoretical Background
### Waveguides of Arbitrary Cross Section Filled by M Metallic Conductors
In addition, \(z_{m}(\tau_{m})=l_{m}(\tau_{m})Z(\tau_{m})\) for \(m=0,\;1,\;2,\ldots,M\). The transformation of the system (30) of first-kind boundary integral equations to a well-conditioned coupled ISLAE of the second kind is performed in a similar manner to that for the isolated waveguide but modified to take account of the additional non-singular coupling terms. Thus, we separate the singular parts from the kernels of the integral equations and then expand their regular parts in their respective Fourier series. First, the extraction of the singular part is expressed by the decomposition, for each index \(s=0,1,2,\ldots,M+1\),  
\[G_{2}(k|p_{m}-q|)=-\frac{i}{4}H_{0}^{(1)}(k|p_{m}-q|). \tag{26}\]1  
As before, the logarithmically singular part has Fourier expansion (see (11))  
\[G_{2}(k|p_{m}-q|)=-\frac{i}{4}H_{0}^{(1)}(k|p_{m}-q|). \tag{26}\]2  
It will be supposed that each contour \(L_{s}\) is such that \(H_{ss}(k;\theta_{s},\tau_{s})\) is smooth and continuously differentiable with respect to \(\theta_{s}\) and \(\tau_{s}\); this allows its expansion in a double Fourier series. \[G_{2}(k|p_{m}-q|)=-\frac{i}{4}H_{0}^{(1)}(k|p_{m}-q|). \tag{26}\]3  
where \(\sum_{p=-\infty}^{\infty}\sum_{n=-\infty}^{\infty}\left(1+p^{2}\right)\left( 1+n^{2}\right)\left|h_{np}^{ss}(k)\right|^{2}<\infty\). The smoothness of the contours and the regularity of the functions \(H_{0}^{(1)}(kR_{ms}(\theta_{s},\tau_{m}))\), for each pair of indices \((m,s)\) with \(m\
eq s\), allows their expansion in double Fourier series  
\[G_{2}(k|p_{m}-q|)=-\frac{i}{4}H_{0}^{(1)}(k|p_{m}-q|). \tag{26}\]4  
Finally, the unknown surface current density on the \(m\)-th conductor may be expanded in a Fourier series:  
\[G_{2}(k|p_{m}-q|)=-\frac{i}{4}H_{0}^{(1)}(k|p_{m}-q|). \tag{26}\]5  
Performing the same steps in the solution obtained in Section 2.1, we arrive at the following coupled system of the linear algebraic equations representing a Fredholm matrix system of the second kind, where the indices \(s\) and \(l\) run over 0, 1, 2, \(\ldots\), \(M\) and 1, 2, \(\ldots\), \(\infty\), respectively:  
\[ \begin{array}{c}X_{l}^{(s)}+\sqrt{l}\left(\kappa_{l,0}^{(s,s)}\right)^{(+,+ )}\xi_{0}^{(s)}+\sqrt{l}\sum_{m=0,m\
eq s}^{\rm M}\left(\chi_{l,0}^{(m,s)} \right)^{(+,+)}\xi_{0}^{(m)}\\ \qquad\qquad+\sum\limits_{p=1}^{\infty}\sqrt{lp}\bigg{\{}\left(\kappa_{l,p}^ {(s,s)}\right)^{(+,+)}\!\!X_{p}^{(s)}-\left(\kappa_{l,p}^{(s,s)}\right)^{(+,- ^{\prime})}\!\!Y_{p}^{(s)}\bigg{\}}\\ +\sum_{m=0,m\
eq s}^{\rm M}\sum_{p=1}^{\infty}\sqrt{lp}\bigg{\{}\left(\chi_{ l,p}^{(m,s)}\right)^{(+,+)}\!\!X_{p}^{(m)}-\left(\chi_{l,p}^{(m,s)}\right)^{(+,- ^{\prime})}\!\!Y_{p}^{(m)}\bigg{\}}=0,\end{array}  \tag{36}\]  
\[ \begin{array}{c}Y_{l}^{(s)}+\sqrt{l}\left(\kappa_{l,0}^{(s,s)}\right)^{(-,+ )}\!\!\xi_{0}^{(s)}+\sqrt{l}\sum_{m=0,m\
eq s}^{\rm M}\!\left(\chi_{l,0}^{(m,s )}\right)^{(-,+)}\!\!\xi_{0}^{(m)}\\ \qquad\qquad+\sum\limits_{p=1}^{\infty}\sqrt{lp}\bigg{\{}\left(\kappa_{l,p}^ {(s,s)}\right)^{(-,+)}\!\!X_{p}^{(s)}-\left(\kappa_{l,p}^{(s,s)}\right)^{(-,- ^{\prime})}\!\!Y_{p}^{(s)}\bigg{\}}\\ +\sum_{m=0,m\
eq s}^{\rm M}\sum_{p=1}^{\infty}\sqrt{lp}\bigg{\{}\left(\chi_{ l,p}^{(m,s)}\right)^{(-,+)}\!\!X_{p}^{(m)}-\left(\chi_{l,p}^{(m,s)}\right)^{(-,- ^{\prime})}\!\!Y_{p}^{(m)}\bigg{\}}=0,\end{array}  \tag{37}\]  
\[ \begin{array}{c}\frac{1}{2}\left(\kappa_{l,0}^{(s,s)}\right)^{(+,+)}\!\!\xi_ {0}^{(s)}+\frac{1}{2}\sum\limits_{m=0,m\
eq s}^{\rm M}\left(\chi_{0,0}^{(m,s)} \right)^{(+,+)}\!\!\xi_{0}^{(m)}\\ \qquad\qquad+\sum\limits_{p=1}^{\infty}\sqrt{p}\bigg{\{}\left(\kappa_{0,p}^{(s,s) }\right)^{(++)}\!\!X_{p}^{(s)}-\left(\kappa_{0,p}^{(s,s)}\right)^{(+,-)}\!\!Y_{p }^{(s)}\bigg{\}}\\ +\sum_{m=0,m\
eq s}^{\rm M}\sum_{p=1}^{\infty}\sqrt{p}\bigg{\{}\left(\chi_{0,p}^ {(m,s)}\right)^{(++)}\!\!X_{p}^{(m)}-\left(\chi_{0,p}^{(m,s)}\right)^{(+,- ^{\prime})}\!\!Y_{p}^{(m)}\bigg{\}}=0,\end{array}  \tag{38}\]where we have used the following notations for the matrix elements:  
\[