= Solution
Write $f=u+iv$. The prescribed complex-linear differential is exactly the <Cauchy-Riemann equations>
$$
u_x=v_y,\qquad u_y=-v_x.
$$
A <holomorphic function> is smooth, as follows locally from its Cauchy integral formula, so its mixed second <partial derivatives> commute. Differentiating the two equations gives
$$
\Delta u=u_{xx}+u_{yy}=v_{yx}-v_{xy}=0,
$$
and, rewriting them as $v_x=-u_y$, $v_y=u_x$,
$$
\Delta v=v_{xx}+v_{yy}=-u_{yx}+u_{xy}=0.
$$
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 $\Delta/2$.
Back to article page
