Let $u(z)$ be harmonic on the annulus $\{a<|z|<b\}$.
- Show that there is a constant $C$ such that $u(z)-C\log|z|$ has a harmonic conjugate on the annulus.
- Show that $C$ is given by $$C=\frac{r}{2\pi}\int_{0}^{2\pi} \frac{\partial u}{\partial r}(re^{i\theta})\,\mathrm{d}\theta.$$
I think the identity can be proved with the polar form of Cauchy-Riemann equations, but I don't know how to prove the existence of $C$.