 Solution

The allowed theorem makes $f^{\prime }$ holomorphic and hence continuous; applying it repeatedly gives continuous derivatives of every order. Since $u_x=\Re f^{\prime }$, $v_x=\Im f^{\prime }$, $u_y=-\Im f^{\prime }$ and $v_y=\Re f^{\prime }$, the real and imaginary parts have continuous second partial derivatives. Differentiate the Cauchy–Riemann equations and commute mixed derivatives:Thus $\boxed{\Delta u=\Delta v=0}$, with the required $C^2$ regularity, proving harmonicity.