 Solution

Write $f=u+iv$. The prescribed complex-linear differential is exactly the Cauchy-Riemann equationsA holomorphic function is smooth, as follows locally from its Cauchy integral formula, so its mixed second partial derivatives commute. Differentiating the two equations givesand, rewriting them as $v_x=-u_y$, $v_y=u_x$,Thus $\boxed{\Delta \Re f=\Delta \Im f=0}$: both components are harmonic functions. Here $\Delta ={\partial }_x^2+{\partial }_y^2$, the sign convention used for the Brownian generator $\Delta /2$.