I have stumbled across the following fact in complex analysis and I was trying to prove it, but didn't get anywhere:
Let $R=\{r<|z|<R\}\subset\mathbb{C}$ where $0<r<R<\infty$ be an annulus in the complex plane and $u$ a harmonic function on $R$. Then there exists a constant $C\in\mathbb{R}$ and a holomorphic function $f$ on $R$ such that $$u(z)=\mathrm{Re}f(z)+C\log|z|$$
The problem arises because $R$ is not simply connected (on simply connected domains this is clearly true for $C=0$ by the CR equations). I just don't see where that $\log$ should come from.
Can somebody help me and provide an easy proof?
Any potentially useful approaches/hints are welcome.