I am a beginner of distribution theory. I would like to find the fundamental solution in the sense of tempered distribution $\mathcal{S}'(\mathbf{R}^3).$(the duality of Schwartz space) \begin{equation} \partial_x E+i\partial_yE+\partial_z^{2}E=\delta, \quad in \quad \mathcal{S}'(\mathbf{R}^3) \end{equation} where $\delta$ is the Dirac distribution in $\mathbf{R}^3.$
My attempt is to apply the Fourier transform on both side of the equation in the sense of $\mathcal{S}'(\mathbf{R}^3),$ we have \begin{equation} (i\xi_1-\xi_2-\xi_3^2)\hat{E}(\xi)=1, \end{equation} where $\hat{E} \in \mathcal{S}'(\mathbf{R}^3).$ Then what can I do next? Can I divide the above formula by $(i\xi_1-\xi_2-\xi_3^2)$, and do the Fourier inverse transform to $(i\xi_1-\xi_2-\xi_3^2)^{-1}$ to get the answer? Can anybody help me? Thank you so much!