Inhomogeneous Wirtinger equation with holomorphic forcing (${\partial }_{\bar{z}}f=g(z)⟺f(z)=\bar{z}g(z)+h(z),g,h\text{ holomorphic}$)

Let $g$ be a holomorphic function on an open set $\Omega \subseteq \mathbb{C}$. Among continuously differentiable functions, all solutions of ${\partial }_{\bar{z}}f=g$ arewhere $h$ is an arbitrary holomorphic function on $\Omega$. The Wirtinger derivatives satisfy ${\partial }_{\bar{z}}\bar{z}=1$ and ${\partial }_{\bar{z}}g=0$, so the product rule verifies the particular solution $\bar{z}g(z)$. Subtracting it from any other solution leaves a continuously differentiable function with vanishing Wirtinger derivative, hence a holomorphic function by the Cauchy-Riemann equations. Conversely adding such a holomorphic function changes no prescribed derivative. For $g(z)=z$ on the complex plane, the solutions are $|z{|}^2+h(z)$ with $h$ any entire function.