## 4 Solution of the Pair of DIEs  
As described in [24], the \(\psi_{i}(\
u)\) functions (which are related to either to the \(\tilde{J}_{5\phi}\) or \(\tilde{M}_{5\rho}\) functions) are expanded in Neumann series of Bessel functions, i.e.,  
\[\psi_{i}(\
u)=\
u^{1-\mu_{i}}\sum_{n=1}^{+\infty}\psi_{n}^{(i)}\mathrm{J}_{2n- 1+\mu_{i}}(\
u)\,; \tag{10}\]  
thus, thanks to the Weber-Schafheitlin integral ([32], 6.574.3), (6) is spontaneously satisfied. The parameters \(\mu_{i}\) in (10) are arbitrary (provided that \(\mu_{i}>0\)), and can be used to guarantee the convergence of the integrals arising in the analysis or to enforce a specific order of singularity of the spatial currents at the edge of the aperture [24]. In fact, all the addends in (10) are the Hankel transforms of terms proportional to  
\[r^{2}\Big{(}1-r^{2}\Big{)}^{\mu_{i}-1}P_{n}^{(1,\mu_{i}-1)}\Big{(}1-2r^{2}\Big{)}\,, \tag{11}\]  
where \(P_{n}^{(\cdot,\cdot)}(\cdot)\) are Jacobi polynomials of order \(n\)[33]. By using (10), (5) becomes  
\[\sum_{n=1}^{+\infty}\psi_{n}^{(i)}\int_{0}^{\infty}\
u^{1-\mu_{i}}\gamma_{i}( \
u)\mathrm{J}_{1}(\
u r)\mathrm{J}_{2n-1+\mu_{i}}(\
u)\;\mathrm{d}\
u=h_{i}(r )\,, \tag{12}\]  
which is valid for \(0\leq r<1\). By Galerkin testing (12), the dual integral equations are reduced to the matrix system  
\[\sum_{n=1}^{+\infty}A_{mn}^{(i)}\psi_{n}^{(i)}=U_{m}^{(i)}\,,\qquad m=1,\dots, \tag{13}\]  
where the matrix elements are provided by  
\[A_{mn}^{(i)}=\int_{0}^{\infty}\
u^{1-2\mu_{i}}\gamma_{i}(\
u)\mathrm{J}_{2m-1 +\mu_{i}}(\
u)\mathrm{J}_{2n-1+\mu_{i}}(\
u)\,\mathrm{d}\
u \tag{14}\]  
while it is easy to show that  
\[U_{m}^{(1)}=-\frac{\mathrm{I}_{0}k_{0}aR}{2}\,\int_{0}^{\infty}\
u^{1-\mu_{i} }\mathrm{J}_{2m-1+\mu_{i}}(\
u)\mathrm{J}_{1}\bigg{(}\
u\frac{R}{a}\bigg{)} \,\frac{\mathrm{e}^{-\mathrm{j}\lambda|h|/a}}{\lambda}\;\mathrm{d}\
u \tag{15}\]  
and  
\[U_{m}^{(2)}=-\frac{\mathrm{I}_{0}R\zeta_{0}}{2}\,\int_{0}^{\infty}\
u^{1-\mu_ {i}}\mathrm{J}_{2m-1+\mu_{i}}(\
u)\mathrm{J}_{1}\bigg{(}\
u\frac{R}{a}\bigg{)} \,\mathrm{e}^{-\mathrm{j}\lambda|h|/a}\;\mathrm{d}\
u\,. \tag{16}\]  
From (8) and (14), it can be immediately derived that to make the integrals convergent it must result \(\mu_{1}>1/2\) and \(\mu_{2}>1\); thus, we can set, e.g., \(\mu_{1}=1\) and \(\mu_{2}=3/2\), and the coefficients in (13) may be obtained as  
\[ \begin{split} A_{mn}^{(1)}&=\int_{0}^{\infty} \frac{\big{(}k_{0}a+2\lambda\hat{Z}_{S}\big{)}}{2\lambda\
u}\,\mathrm{J}_{2m}( \
u)\mathrm{J}_{2n}(\
u)\,\mathrm{d}\
u\,,\\ A_{mn}^{(2)}&=\int_{0}^{\infty}\frac{\big{(}\lambda+2 k_{0}a\hat{Y}_{S}\big{)}}{2k_{0}a\
u^{2}}\,\mathrm{J}_{2m+1/2}(\
u)\mathrm{J}_{2n+1/2 }(\
u)\,\mathrm{d}\
u\,,\end{split}  \tag{17}\]  
i.e.,  
\[ \begin{split} A_{mn}^{(1)}&=\frac{k_{0}a}{2}\,\int_ {0}^{\infty}\frac{\mathrm{J}_{2m}(\
u)\mathrm{J}_{2n}(\
u)}{\lambda\
u}\; \mathrm{d}\
u+\frac{\hat{Z}_{S}}{4m}\,\delta_{mn}\,,\\ A_{mn}^{(2)}&=\frac{1}{2k_{0}a}\,\int_{0}^{\infty} \frac{\big{(}\lambda+2k_{0}a\hat{Y}_{S}\big{)}}{\
u^{2}}\,\mathrm{J}_{2m+1/2}( \
u)\mathrm{J}_{2n+1/2}(\
u)\,\mathrm{d}\
u\end{split}  \tag{18}\]  
and the known terms in (13) as  
\[ \begin{split} U_{m}^{(1)}&=-\frac{\mathrm{I}_{0}k_{0 }aR}{2}\,\int_{0}^{\infty}\mathrm{J}_{2m}(\
u)\mathrm{J}_{1}\bigg{(}\
u\frac{R }{a}\bigg{)}\,\frac{\mathrm{e}^{-\mathrm{j}\lambda|h|/a}}{\lambda}\;\mathrm{d} \
u\,,\\ U_{m}^{(2)}&=-\frac{\mathrm{I}_{0}R\zeta_{0}}{2}\, \int_{0}^{\infty}\frac{\mathrm{J}_{2m+1/2}(\
u)}{\sqrt{\
u}}\,\mathrm{J}_{1} \bigg{(}\
u\frac{R}{a}\bigg{)}\,\mathrm{e}^{-\mathrm{j}\lambda|h|/a}\;\mathrm{ d}\
u\,.\end{split}