is there a harmonic function such that ${\partial f \over \partial r}$ isn't harmonic?

Question

I've tried bunch of imaginary and real parts of holomorphic functions but it's always harmonic. and my teacher assured me that ${\partial f \over \partial r}$ isn't necessarily harmonic.

Answer 1

Plenty such examples.

$$ u(x,y)=x=r\cos \vartheta=u(r,\vartheta) $$ is harmonic, but $$ \frac{\partial}{\partial r}u(r,\vartheta)=\cos \vartheta=\frac{x}{\sqrt{x^2+y^2}} $$ is not harmonic.

Answer 2

The reason why it's not true in general is that the laplacian in polar coordinates does not have constant coefficients, so does not commute with $\frac{\partial}{\partial r}$ $$\Delta f= \frac{\partial^2 f}{\partial r^2} + \frac{1}{r}\frac{\partial f}{\partial r} + \frac{1}{r^2}\frac{\partial^2 f}{\partial \theta^2}$$

An important example of harmonic function is the real part of $z\mapsto \log z$, that is, $\log r$. You can check that $\frac{1}{r}$ is not harmonic.