\(u=2\ln|z|=\operatorname{Re}(2\log z)\), whose local conjugate is \(2\arg z\); going once around \(0\) it changes by \(4\pi\). The domain is not simply connected.
A real function \(u\) of two variables, with continuous second partial derivatives on an open set \(U\) of the plane, is harmonic if its Laplacian \(\Delta u\) is zero throughout \(U\).