Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2010 / Analysis / 4 / iii / Solution

Write $f=u+iv$ and evaluate all derivatives of $\psi$ at $(u,v)$. The real chain rule givesThe Cauchy–Riemann equations give equal squared lengths $u_x^2+u_y^2=v_x^2+v_y^2=|f^{\prime }{|}^2$ and zero cross term $u_x{v}_x+u_y{v}_y=0$. Part (ii) removes the last two terms. ThereforeThe composition is $C^2$, so it is harmonic. No assumption that $f^{\prime }$ is nonzero is required.