Let $u(x,y)$ and $v(x,y)$ be a pair of conjugate harmonic functions, and let $x(u, v)$ be a harmonic function of the variables $u$ and $v$.
Prove that $\chi [u(x,y),v(x,y)]$ is a harmonic function of the variables $x$ and $y$.
I know that $u$ and $v$ satisfies the Cauchy Riemann ecuations, $\frac{\partial^2u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0$ (similar for $v$), and exist and are continuous $\frac{\partial^2 u}{\partial x^2}, \frac{\partial ^2 u}{\partial x \partial y}, \frac{\partial^2 u}{\partial y \partial x}, \frac{\partial^2u}{\partial y^2}$ (similar for $v$) and analogously for $x(u,v)$.
I guess I have to prove that the same occurs for $\chi$, but I can't see the relation between the partial derivatives of $ \chi $ and those of $ x (u, v) $.
I hope you can help me, thank you.
Let $f = u+iv$ so that $f$ is analytic. $\chi$ is harmonic so it is locally the real part of an analytic function $g$. Compositions of analytic functions are analytic and hence $\chi\circ f = \mathrm{Re}g\circ f$ is the real part of an analytic function, hence harmonic