Dbar equation (${\partial }_{\bar{z}}F=q$)

A prescribed inhomogeneous Wirtinger derivative equation. The distributional identity ${\partial }_{\bar{z}}(1/(\pi z))={\delta }_0$ gives its whole-plane right inverse through the Cauchy-Green operator. Boundary or decay conditions select a solution from the freedom to add a holomorphic function.

A dbar equation in a complex spectral parameter, rather than the original spatial coordinate. In spectral reconstruction, its right-hand side is transformed data. A Cauchy-Pompeiu formula in the spectral plane reconstructs the auxiliary function; an asymptotic coefficient or the original differential equation then recovers the spatial field.