Laplacian identity for the squared modulus of a holomorphic function
= Laplacian identity for the squared modulus of a holomorphic function
{c}
{title2=$\Delta|f|^2=4|f'|^2$}
Write a <holomorphic function> as $f=u+iv$. The <Cauchy-Riemann equations> make $u,v$ <harmonic functions> and give $u_x=v_y$, $u_y=-v_x$. The <product rule> therefore gives $\Delta(u^2+v^2)=2(|\nabla u|^2+|\nabla v|^2)=4|f'|^2$. Thus its squared <modulus> is <subharmonic> and is harmonic only where $f'$ vanishes.