- 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 $$ C=\frac{r}{2 \pi} \int_{0}^{2 \pi} \frac{\partial u}{\partial r}\left(r e^{i \theta}\right) d \theta $$
I know $u(z)-\log|z|$ is harmonic on $D=\{a<|z|<b\}\setminus(-b,-a)$, so it has a harmonic conjugate $o$ on $D$ because it is a simply connected domains. But $\{a<|z|<b\}$ is not a simply connected domains. So we should prove $Pdx+Qdy$ is exact on $\{a<|z|<b\}$. I can't prove it.
This problem comes from Gamelin's complex analysis.
