Let $u: U \subset \Bbb C \to \Bbb R$ be harmonic. We define $f:=u_x -iu_y$. I wanna show that $f$ satisfies the Cauchy-Riemann equations but I am not sure.
I should proof for the first equation that $(u_x)_x=-(u_y)_y$, right? The $i$ is skipped, right? We have that $u$ is harmonic and hence $\bigtriangleup u= \frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0$. But this means that $(u_x)_x=-(u_y)_y$ and this proofed the first equation.
The second equation would be $(u_x)_y=(u_y)_x$. But this follows from schwarz lemma not?
Hence the Cauchy-Riemann equations hold for $f$. Is this right?