for Complex-Valued Conductivities {#sec:FranDbar}
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The methods in this section can reconstruct real or complex-valued admittivities $\gamma(z)=\sigma(z)+i\omega\epsilon(z)$. We briefly review the methods developed in \[[@Francini2000; @Hamilton2012; @Herrera2015]\].

First, equation is transformed to a first-order elliptic system of equations $$\label{eq:DQ}
\left[-D+ Q(z)\right]\Psi(z,k)=0, \quad z,k\in{{\mathbb C}},$$ where $$D=\left[\begin{array}{cc}
\bar\partial_z & 0\\
0 &\partial_z
\end{array}\right], \quad 
Q(z)=\left[\begin{array}{cc}
0 & -\frac{1}{2} \partial_z \log \gamma\\
-\frac{1}{2} \bar\partial_z \log \gamma & 0\\
\end{array}\right], \quad \text{and } \Psi(z,k)=\gamma^{1/2}(z)\left[\begin{array}{cc}
\operatorname{\operatorname{\partial}_z}u_1& \operatorname{\operatorname{\partial}_z}u_2\\
\operatorname{{\bar{\partial}}_z}u_1 & \operatorname{{\bar{\partial}}_z}u_2\\
\end{array}\right],$$ where $u_1\sim \frac{e^{ikz}}{ik}$ and $u_2\sim \frac{e^{-ik\bar{z}}}{-ik}$ are CGO solutions to the admittivity equation , whose existence was established in \[[@Hamilton2012]\].

Functions defined by $M(z,k) \equiv \Psi(z,k)\left[\begin{array}{cc}
e^{-ikz} &0\\
0 & e^{ik\bar{z}}\\
\end{array}\right]$ are CGO solutions to a system of D-bar equations in the transform variable $k$ $$\label{eq:Fran_dbark}
\overline{\partial}_k M(z,k)=M(z,\bar{k})\left[\begin{array}{cc}
e(z,\bar{k}) & 0 \\
0 & e(z,-k)
\end{array}\right]S(k),$$ where $S(k)$ is now a matrix of scattering data defined by $$\label{eq:Fran_scatDom}
S(k)=\left[\begin{array}{cc}
0 & \frac{i}{\pi}\int_{{{\mathbb R}}^2} e(z,-\bar{k})Q_{12}(z)M_{22}(z,k)dz\\
-\frac{i}{\pi}\int_{{{\mathbb R}}^2} e(z,k)Q_{21}(z)M_{11}(z,k)dz & 0\\
\end{array}\right].$$ Since $\gamma\equiv 1$ in ${{\mathbb R}}^2\setminus \Omega$, the matrix potential $Q$ has compact support, and integration by parts in results in $$\begin{aligned}
S_{21}(k) &=& -\frac{i}{2\pi}\int_{\operatorname{\partial\Omega}}e^{i\bar{k}\bar{z}}\Psi_{21}(z,k)\overline{\nu(z)}\;ds(z)\label{eq:Fran_PsiScat21} \\
S_{12}(k) &=& \frac{i}{2\pi}\int_{\operatorname{\partial\Omega}}e^{-i\bar{k}{z}}\Psi_{12}(z,k){\nu(z)}\;ds(z),   \label{eq:Fran_PsiScat21}\end{aligned}$$ where $\nu$ denotes the unit outward normal vector to $\operatorname{\partial\Omega}$.

The admittivity 