= Inhomogeneous Wirtinger equation with holomorphic forcing
{title2=$\partial_{\bar z}f=g(z)\iff f(z)=\bar z\,g(z)+h(z),\quad g,h\text{ holomorphic}$}
Let $g$ be a <holomorphic function> on an <open set> $\Omega\subseteq\mathbb C$. Among <continuously differentiable functions>, all solutions of $\partial_{\bar z}f=g$ are
$$
f(z)=\bar z\,g(z)+h(z),
$$
where $h$ 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>.
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