I'm trying to learn a bit of complex analysis, and this idea has got me stuck.
I would like to show that, for $u$ a function of a complex variable $z$, that $u(z)$ and $u(\bar{z})$ are simultaneously harmonic.
I try writing $u(z)=a(z)+ib(z)$. Assuming $u(z)$ is harmonic, $ \frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0. $ Also, I think $ \frac{\partial^2 u}{\partial x^2}=\frac{\partial^2 a}{\partial x^2}+i\frac{\partial^2 b}{\partial x^2}, \qquad \frac{\partial^2 u}{\partial y^2}=\frac{\partial^2 a}{\partial y^2}+i\frac{\partial^2 b}{\partial y^2}. $ I don't understand how to use this to show $u(z)$ and $u(\bar{z})$ are simultaneously harmonic. Aren't these the same function $u$? Shouldn't that be independent of whether you plug in $z$ or $\bar{z}$? Thanks.