# Analytically Regularized Evaluation of the Coupling of Planar Concentric Conducting Rings
## 2 Statement of the Problem
### MoM Solution: Expansion  
Equation (6) can be solved numerically by means of MoM. However, caution has to be paid when resorting to numerical methods, as their accuracy and effectiveness, or even their convergence, cannot be taken for granted at all. MAP can allow it to achieve all three mentioned attractive features, by applying a Galerkin discretization scheme, with a suitable selection of expansion and projection functions, to recast the integral equation as a Fredholm second-kind matrix operator equation. In particular, such an approach properly works, i.e., Fredholm theory can be applied, even when the most singular part of the obtained matrix operator is not diagonal but simply invertible with a continuous two-side inverse and the remaining part is a compact operator [26]. As a matter of fact, it has been shown that such a goal is achieved when using expansion functions factorizing the correct edge behavior of the unknowns in the spatial domain. Furthermore, in order for the method to also be effective, it is desirable to be able to perform the transform of the expansion functions analytically. All mentioned features can be found in the following functions:  
\[\psi_{m}(\rho)=\frac{4\rho p(\rho)}{\pi ab}T_{n}\left(\frac{\rho^{2}+r_{i}r_{ o}}{2\rho a}\right)T_{m}\left(1-2\frac{\rho^{2}-r_{i}^{2}}{r_{o}^{2}-r_{i}^{2}} \right)\quad with\quad\rho\in[r_{i},r_{o}] \tag{7}\]  
\[\chi_{m}(\rho)=\frac{4\rho}{\pi p(\rho)ab}T_{n}\left(\frac{\rho^{2}+r_{i}r_{ o}}{2\rho a}\right)T_{m}\left(1-2\frac{\rho^{2}-r_{i}^{2}}{r_{o}^{2}-r_{i}^{2}} \right)\quad with\quad\rho\in[r_{i},r_{o}] \tag{8}\]  
where \(T_{h}(\cdot)\) is the Chebychev polynomial of first kind and order \(h\), and the weighting function \(p(\rho)=\sqrt{(\rho^{2}-r_{i}^{2})(r_{o}^{2}-\rho^{2})}\) factorizes the correct edge behavior of the current components according to Meixner conditions [48]. Thus, (7) and (8) can be used as expansion functions of radial and azimuthal components of the induced current density, respectively, as shown later in expression (13). In particular, it is worth noting that in (7) and (8) the first Chebychev polynomial, which is a continuous, smooth, functional, and depends on the azimuthal index, but is independent of \(m\) and the weighting function \(p(\rho)\)Consequently, when summing over \(m\) it is factorized and the reconstruction of the smooth part of the current density is left to the Chebychev polynomials of order \(m\). Furthermore, the Hankel transform of functions (7) and (8) is analytical; indeed, the scalar Hankel transform of \(\chi_{m}(\rho)\) is known and can be written as [47]  
\[\int_{r_{i}}^{r_{o}}\rho\chi_{m}(\rho)J_{n}(w\rho)d\rho=J_{n+m}(wa)J_{m}(wb) \equiv F_{m}(w) \tag{9}\]  
where \(a=(r_{i}+r_{o})/2\) and \(b=(r_{o}-r_{i})/2\). Since \(\chi_{m}(\rho)\) is suitable as an expansion function of the azimuthal component of the current density, resembling notation (1), we can consider the quantity \( \begin{bmatrix}0\\ \chi_{m}(\rho)\end{bmatrix} \), for which it is easy to show that  
\[VHT \begin{bmatrix}0\\ \chi_{m}(\rho)\end{bmatrix} = \begin{bmatrix}\frac{n}{w}\\ \frac{d}{dw}\end{bmatrix} F_{m}(w) \tag{10}\]  
Moreover, using the recurrence relations of Chebychev polynomials, after some algebraic manipulations it is possible to demonstrate the following recurrence relationship  
\[\int_{r_{i}}^{r_{o}}\rho\psi_{m}(\rho)J_{n}(w\rho)d\rho=2F_{m}(w)-F_{m-2}(w)-F_ {m+2}(w)\equiv G_{m}(w) \tag{11}\]  
which also defines the scalar Hankel transform of \(\psi_{m}(\rho)\). Similar to what was conducted before, we can consider the quantity \( \begin{bmatrix}\psi_{m}(\rho)\\ 0\end{bmatrix} \) and calculate the corresponding VHT as  
\[VHT \begin{bmatrix}\psi_{m}(\rho)\\ 0\end{bmatrix} = \begin{bmatrix}\frac{d}{dw}\\ \frac{n}{w}\end{bmatrix} G_{m}(w) \tag{12}\]  
Finally, it is worth noting that recurrence relation (11) can be very helpful in the evaluation of the scattering matrix in order to reduce the overall number of integrals to be calculated numerically.