I'm reading something, and don't understand why a certain equality comes up.
Suppose $u:\mathbb{C}\setminus\{0\}\to\mathbb{R}$ given by $u(z)=\ln(|z|^2)$ is a harmonic function. We want to see if $u$ has a harmonic conjugate $v$. If it does, then $f=u+iv$ is holomorphic.
But then I read that since the differential of $f$ is complex linear, then $ \frac{\partial}{\partial\theta}f(re^{i\theta})=i\frac{\partial}{\partial r}f(re^{i\theta}). $
I don't follow this. Even writing it out with the chain rule in terms of $u$ and $v$ doesn't make it clear to me. Why does this equality follow? Thanks.