Wirtinger derivatives

Remarks

Example: for $f(z) = \abs{z}^2 = x^2 + y^2$, one gets $\partial f / \partial z = \frac{1}{2}(2x - 2iy) = \bar z$ and $\partial f / \partial \bar z = \frac{1}{2}(2x + 2iy) = z$. Non-example of holomorphy: since $\partial f / \partial \bar z = z \neq 0$ for $z \neq 0$, the function $\abs{z}^2$ is not holomorphic on any open set. For $f$ of class $C^1$ on $U$, the equation $\partial f / \partial \bar z = 0$ on $U$ is equivalent to the Cauchy–Riemann equations, so it holds exactly when $f$ is holomorphic on $U$; then $\partial f / \partial z = f'$. The operators obey the rules $\partial z / \partial z = 1$, $\partial \bar z / \partial z = 0$, $\partial z / \partial \bar z = 0$ and $\partial \bar z / \partial \bar z = 1$, together with linearity and the product rule. The Laplacian is $\Delta f = 4 \, \dfrac{\partial^2 f}{\partial z \, \partial \bar z}$ for $f$ of class $C^2$. The notations $\partial_z f$ and $\partial_{\bar z} f$ are also common.