decouples into two systems of two equations, each which can be solved using the integral form $$\label{eq:Fran_dbark_convs}
\begin{array}{c}
\left\{\begin{array}{lcl}
1 & =& M_{11}(z,k) - \frac{1}{\pi k}\ast\left[M_{12}(z,\bar{k})e(z,-k)\operatorname{S_{21}^{\mbox{\tiny{\textbf{exp}}}}}(k)\right]\\
0 & =& M_{12}(z,k) - \frac{1}{\pi k}\ast\left[M_{11}(z,\bar{k})e(z,\bar{k})\operatorname{S_{12}^{\mbox{\tiny{\textbf{exp}}}}}(k)\right]
\end{array}\right.\\
\\
\left\{\begin{array}{lcl}
1 & =& M_{22}(z,k) - \frac{1}{\pi k}\ast\left[M_{21}(z,\bar{k})e(z,\bar{k})\operatorname{S_{12}^{\mbox{\tiny{\textbf{exp}}}}}(k)\right]\\
0 & =& M_{21}(z,k) - \frac{1}{\pi k}\ast\left[M_{22}(z,\bar{k})e(z,-k)\operatorname{S_{21}^{\mbox{\tiny{\textbf{exp}}}}}(k)\right],
\end{array}\right.\\
\end{array}$$ where $\ast$ denotes convolution in $k$ over the disc of radius $R$, since $\operatorname{S^{\mbox{\tiny{\textbf{exp}}}}}$ has compact support in $|k|\leq R$, and we have used the fundamental solution $\frac{1}{\pi k}$ for the $\operatorname{{\bar{\partial}}_k}$ operator as well as the asymptotic condition $M(z,k)\sim \left[\begin{array}{cc} 1 & 0\\ 0 & 1\\ \end{array}\right]$ for large $|k|$ or $|z|$. The solution can be obtained using a modified version of Vainikko’s method \[[@Vainikko2000]\] for solving integral equations with weakly singular kernels. The convolutions can be implemented using two-dimensional [FFT]{}s on a uniform $k$-grid of size $(2^{N}+1)\operatorname{\times}(2^{N}+1)$ with uniform step size $h_k$ as $$\frac{1}{\pi k}\ast f(k)=h_k^2\;IFFT\left[FFT\left(\frac{1}{\pi k}\right)\;FFT\left(f(k)\right)\right].$$ The system in can be solved using matrix-free [GMRES]{} for each desired $z\in\Omega$. Note that this step is trivially parallelizable in $z$, and that after each solution of only the $k=0$ entry is required to recover the admittivity $\operatorname{\gamma^{\mbox{\tiny{\textbf{exp}}}}}(z)$.

The potentials $\operatorname{Q_{12}^{\mbox{\tiny{\textbf{exp}}}}}$ and $\operatorname{Q_{21}^{\mbox{\tiny{\textbf{exp}}}}}$ can be computed from , 