Wirtinger derivatives (${\partial }_z,{\partial }_{\bar{z}}$)

For $z=x+iy$, the Wirtinger derivatives are ${\partial }_z=({\partial }_x-i{\partial }_y)/2$ and ${\partial }_{\bar{z}}=({\partial }_x+i{\partial }_y)/2$. The Cauchy-Riemann equations become ${\partial }_{\bar{z}}F=0$ for a holomorphic function $F$, and $4{\partial }_z{\partial }_{\bar{z}}=\Delta$. Thus $u_z$ is holomorphic whenever $u$ is a twice continuously differentiable harmonic function.

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.