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**romsek** just show that $\nabla^2 \varphi(x,y) = 0$

$\varphi(x,y)=uv$

$\varphi_x=\dfrac {\partial}{\partial x} \varphi = u_x v + u v_x$

$\varphi_y=\dfrac {\partial}{\partial y} \varphi = u_y v + u v_y$

$\varphi_{xx}=\dfrac {\partial}{\partial x}\left(u_x v + u v_x\right)=u_{xx}v + 2 u_x v_x + u v_{xx}$

$\varphi_{yy}=\dfrac {\partial}{\partial y}\left( u_y v + u v_y \right) = u_{yy} v + 2 u_y v_y + u v_{yy}$

$\varphi_{xx}+\varphi_{yy}=v(u_{xx}+u_{yy}) + u(v_{xx}+ v_{yy}) + 2(u_x v_x + u_y v_y) = 0 + 0 + u_x (-u_y) + u_y (u_x)=0$

the Laplacians of u and v in each 0 as they are each harmonic functions.

the last equality is from the CR equations of u and v which are satisfied as they are harmonic conjugates of one another.