 Solution

A real harmonic function on a planar open set is a twice continuously differentiable function satisfying $f_{xx}+f_{yy}=0$. The ordered pair $(f,g)$ consists of harmonic conjugates when $f+ig$ is holomorphic; the corresponding Cauchy-Riemann equations are $f_x=g_y$ and $f_y=-g_x$.

To prove composition of harmonic conjugate pairs, denote $P=p(u,v)$ and $Q=q(u,v)$, and assume the inner pair's image lies in the outer pair's domain. Their equations are $u_x=v_y$, $u_y=-v_x$, $p_u=q_v$ and $p_v=-q_u$. The chain rule givesThus the composite satisfies the Cauchy-Riemann equations. Its components are $C^2$, and differentiating gives $P_{xx}+P_{yy}=Q_{yx}-Q_{xy}=0$ and similarly $Q_{xx}+Q_{yy}=0$. The composite pair is harmonic conjugate, including when the inner map has a critical point.