## 3 Pair of DEs through Mitzner GBCs  
The finite thickness of the screen makes the analysis more complex with respect to the infinitesimally-thin structure case. However, if the thickness \(d\) is sufficiently small (in particular, \(k_{0}d\ll 1\), where \(k_{0}\) is the free-space wavenumber), it is possible to shrink the thickness to zero (reducing the screen to an infinitely thin surface) and adopt the Mitzner generalized boundary conditions (GBCs) [25]. The GBCs link the tangential electric and magnetic fields at the surface interfaces as  
\[\frac{1}{2}\left[E_{\phi}\big{(}\rho,z=0^{+}\big{)}+E_{\phi}\big{(}\rho,z=0^{- }\big{)}\right]=\zeta_{0}\hat{Z}_{S}\big{(}\mathbf{u}_{z}\times\mathbf{u}_{ \rho}\big{)}\big{[}H_{\rho}\big{(}\rho,z=0^{+}\big{)}-H_{\rho}\big{(}\rho,z=0^ {-}\big{)}\big{]} \tag{1}\]  
and  
\[\frac{1}{2}\left[H_{\rho}\big{(}\rho,z=0^{+}\big{)}+H_{\rho}\big{(}\rho,z=0^{- }\big{)}\right]=-\frac{\hat{Y}_{S}}{\zeta_{0}}\left(\mathbf{u}_{z}\times \mathbf{u}_{\phi}\right)\left[E_{\phi}\big{(}\rho,z=0^{+}\big{)}-E_{\phi} \big{(}\rho,z=0^{-}\big{)}\right], \tag{2}\]  
where \(\zeta_{0}\) is the free-space impedance. Such GBCs have recently been used successfully in different scattering problems [16; 17; 27; 29; 30]. The coefficients \(\hat{Z}_{S}\) and \(\hat{Y}_{S}\) are related to the thickness and the electromagnetic parameters of the screen, and different expressions have been proposed in the literature. In his seminal paper [25], [PERSON] derived  
\[\hat{Z}_{SM}=\mathrm{j}\frac{\zeta_{\mathrm{cr}}}{2}\ \cot\!\left( \frac{k_{\mathrm{c}}d}{2}\ \right), \tag{3}\] \[\hat{Y}_{SM}=\mathrm{j}\frac{1}{2\zeta_{\mathrm{cr}}}\ \cot\!\left( \frac{k_{\mathrm{c}}d}{2}\ \right)\]  
with  
\[k_{\mathrm{c}}=k_{0}\sqrt{\mu_{\mathrm{r}}\epsilon_{\mathrm{cr} }}, \tag{4}\] \[\zeta_{\mathrm{cr}}=\sqrt{\frac{\mu_{\mathrm{r}}}{\epsilon_{ \mathrm{cr}}}}\ \,\]  
where \(\epsilon_{\mathrm{cr}}=1-\mathrm{j}\sigma/(\omega\epsilon_{0})\) indicates the relative complex permittivity. It is worth noting that the condition \(k_{0}d\ll 1\) is not a limiting factor in all the practical configurations, where the thickness \(d\) of the screen is much smaller than the operating wavelength in air up to the microwave range. Figure 1: A circular disk of thickness \(d\) and radius \(a\) characterized by a conductivity \(\sigma\) and relative magnetic permeability \(\mu_{\mathrm{r}}\), placed on the plane \(z=0\), in the presence of a current loop of radius \(R\) and constant current \(l_{0}\) placed at \(z=h\) and coaxial with the disk. The GBCs in (1) and (2) define a combined electrically resistive/magnetically conductive sheet with infinitesimal thickness, which supports both electric and magnetic surface currents that are related to the discontinuities across the sheet of the tangential magnetic and electric fields, respectively. Because any plane containing the vertical \(z\)-axis is an odd symmetry plane, the electric surface current density \(\mathbf{J}_{\mathrm{s}}\) is azimuthally directed, whereas the magnetic current density \(\mathbf{M}_{\mathrm{s}}\) is radially directed. Following the same reasoning illustrated in [31], two sets of DIEs can be obtained in the canonical form  
\[\int_{0}^{\infty}\gamma_{i}(\
u)\psi_{i}(\
u)\mathrm{J}_{1}(\
u r)\,\mathrm{d} \
u=h_{i}(r),\quad 0\leq r<1 \tag{5}\]  
\[\int_{0}^{\infty}\psi_{i}(\
u)\mathrm{J}_{1}(\
u r)\,\mathrm{d}\
u=0,\quad r>1, \tag{6}\]  
where \(r=\rho/a,\mathrm{J}_{1}(\cdot)\) is the Bessel function of order 1.