Cauchy–Riemann operator

Remarks

Writing $f = u + iv$ with $u, v$ real, $\partial f / \partial \bar z = 0$ is equivalent to $u_x = v_y$ and $u_y = -v_x$, the Cauchy–Riemann equations. So a continuously differentiable function $f$ on $\Omega$ is holomorphic if and only if $\partial f / \partial \bar z = 0$ on $\Omega$. Example: $\partial z / \partial \bar z = \tfrac12 (1 + i \cdot i) = 0$, so $z$ is in the kernel. Non-example: $\partial \bar z / \partial \bar z = \tfrac12 (1 + i \cdot (-i)) = 1 \ne 0$, so $\bar z$ is not holomorphic. The companion operator $\partial / \partial z = \tfrac12 (\partial_x - i \partial_y)$ satisfies $4 \, \partial_z \partial_{\bar z} = \Delta$. In distributions on $\C$, $\partial_{\bar z} \big( 1/(\pi z) \big) = \delta_0$, the Dirac measure at $0$.