## 2 Formulation of the Problem
By introducing the following auxiliary unknown:  
\[\underline{\tilde{\mathbf{J}}}_{e}^{(n)}(w)=\underline{\widetilde{\mathbf{G}}} (w)\underline{\tilde{\mathbf{J}}}_{i}^{(n)}(w)+\Big{(}\underline{\widetilde{ \mathbf{G}}}(w)-R_{\underline{1}}\underline{\tilde{\mathbf{J}}}_{e}^{(n)}(w)+ \underline{\tilde{\mathbf{E}}}_{t}^{\text{inc}(n)}(w,0), \tag{9}\]  
where  
\[\underline{\tilde{\mathbf{E}}}_{t}^{\text{inc}(n)}(w,0)=-\text{j}^{n+1}e^{- \text{j}n\phi_{0}}\frac{\delta(w-k_{0}s_{\theta_{0}})}{w}\underline{\mathbf{E}}_ {0t} \tag{10}\]  
denotes the VHT\({}_{n}\) of \(\underline{\mathbf{E}}_{t}^{\text{inc}(n)}(\rho,0)\), the following alternative expressions for the equations in (7) can be readily obtained:\[\int\limits_{0}^{+\infty}\underline{\mathbf{H}}^{(n)}(w\rho)\bigg{\{} \bigg{[}\widetilde{\underline{\underline{\mathbf{G}}}}(w)-R_{i\underline{ \mathbf{I}}}-\widetilde{\underline{\mathbf{G}}}(w)\Big{(}\widetilde{\underline {\underline{\mathbf{G}}}}(w)-R_{e\underline{\mathbf{I}}}\Big{)}^{-1} \widetilde{\underline{\mathbf{G}}}(w)\bigg{]}\underline{\underline{\mathbf{I} }}_{i}^{(n)}(w)+\underline{\widetilde{\mathbf{G}}}(w)\Big{(}\widetilde{ \underline{\underline{\mathbf{G}}}}(w)-R_{e\underline{\mathbf{I}}}\Big{)}^{-1} \widetilde{\underline{\mathbf{I}}}_{e}^{(n)}(w)\bigg{\}}wdw\] (11a) \[=-\underline{\mathbf{E}}_{t}^{tr(n)}(\rho,0)\text{ for }\rho<a,\] \[\int\limits_{0}^{+\infty}\underline{\mathbf{H}}^{(n)}(w\rho) \underline{\tilde{\mathbf{J}}}_{i}^{(n)}(w)wdw=\underline{0}\text{ for }\rho>a,\]0 (11b) \[\int\limits_{0}^{+\infty}\underline{\mathbf{H}}^{(n)}(w\rho) \underline{\tilde{\mathbf{J}}}_{i}^{(n)}(w)wdw=\underline{0}\text{ for }\rho>a,\]1 (11c) \[\int\limits_{0}^{+\infty}\underline{\mathbf{H}}^{(n)}(w\rho) \underline{\tilde{\mathbf{J}}}_{i}^{(n)}(w)wdw=\underline{0}\text{ for }\rho>a,\]2 (11d) where \[\int\limits_{0}^{+\infty}\underline{\mathbf{H}}^{(n)}(w\rho) \underline{\tilde{\mathbf{J}}}_{i}^{(n)}(w)wdw=\underline{0}\text{ for }\rho>a,\]3 (12) is the \(n\)-th azimuthal harmonic of the transverse component of the field transmitted by the homogeneous resistive plane of resistivity \[\int\limits_{0}^{+\infty}\underline{\mathbf{H}}^{(n)}(w\rho) \underline{\tilde{\mathbf{J}}}_{i}^{(n)}(w)wdw=\underline{0}\text{ for }\rho>a,\]4. By substituting (11d) in (11a), we obtain \(\widetilde{\underline{\mathbf{I}}}_{e}^{(n)}(\rho)=R_{i\underline{\mathbf{I}}} _{i}^{(n)}(\rho)\) for \(\rho<a\). On the other hand, from (11b) and (11c), we immediately conclude that \(\widetilde{\underline{\mathbf{I}}}_{e}^{(n)}(\rho)=\underline{\underline{ \mathbf{I}}}_{i}^{(n)}(\rho)=\underline{0}\) for \(\rho>a\). Hence,  
\[\int\limits_{0}^{+\infty}\underline{\mathbf{H}}^{(n)}(w\rho) \underline{\tilde{\mathbf{J}}}_{i}^{(n)}(w)wdw=\underline{0}\text{ for }\rho>a,\]5  
independently of \(\rho\), and Equation (11) is reduced to the following dual integral equations:  
\[\int\limits_{0}^{+\infty}\underline{\mathbf{H}}^{(n)}(w\rho) \underline{\tilde{\mathbf{J}}}_{i}^{(n)}(w)wdw=\underline{0}\text{ for }\rho>a,\]6 \[\int\limits_{0}^{+\infty}\underline{\mathbf{H}}^{(n)}(w\rho) \underline{\tilde{\mathbf{J}}}_{i}^{(n)}(w)wdw=\underline{0}\text{ for }\rho>a,\]7  
The functional space to which \(\underline{\underline{\mathbf{I}}}_{i}^{(n)}(\rho)\) belongs can be readily established. Indeed, \(\underline{\mathbf{J}}_{i}^{(n)}(\rho)\) vanishes for \(\rho>a\) and is well-behaved in \(0<\rho<a\), because the sources are off the scatterer surface. Moreover, the physical behavior prescribed for the fields at the discontinuity between the two involved resistive media [49] allows the conclusion that \(\underline{\mathbf{J}}_{i}^{(n)}(\rho)\overset{\rho\to a}{\sim}\text{O}(1)\), whereas, according to the Fourier series expansion properties, \(\underline{\mathbf{J}}_{i}^{(n)}(\rho)\overset{\rho\to 0}{\sim}\rho^{||n|-1|}\).