= Solution
For real-direction increments, <complex differentiability> gives $f'=u_x+iv_x$. For imaginary-direction increments it gives $f'=(u_y+iv_y)/i=v_y-iu_y$. Equating these expressions gives the <Cauchy-Riemann equations>
$$
\boxed{u_x=v_y,\qquad u_y=-v_x.}
$$
A <holomorphic function> is smooth, so mixed <partial derivatives> commute. Differentiating these equations gives
$$
u_{xx}+u_{yy}=v_{yx}-v_{xy}=0,\qquad v_{xx}+v_{yy}=-u_{yx}+u_{xy}=0.
$$
Thus both real and imaginary parts are <harmonic functions>:
$$
\boxed{\nabla^2u=\nabla^2v=0.}
$$
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