I'm stuck with this exercise, I don't know how characterize the harmonic functions of the exercise. I'd appreciate your help. Thank you.
Let $G=\mathbb C\setminus\{(-\infty,0]\}$. Find all the harmonics functions such that are constants on the rays; a ray is a set $\{z\in \mathbb C: \arg z=c\}$.
Let $u$ be a Harmonic function on $G$. It is clear that $G$ is simple connected, since its complement in $\mathbb C\cup\{\infty\}$ is connected. There exists the Harmonic conjugate $v$ of $u$ on $G$. That is, $f=u+iv$ is analytic on $G$. Let $R_\theta= \{z\in G: \arg z= \theta\}$, then by hypothesis the real part of $f$ is constant on $R_\theta$. The idea is to use the C-R eqn.
A harmonic function $u$ satisfies Laplace equation $$\frac{\partial^2 u}{\partial r^2} +\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^2}\frac{\partial^2 u}{\partial \theta ^2}=0.\tag{1}$$ If $u$ is constant on every ray, then $$ \frac{\partial^2 u}{\partial r^2}=0, \quad \frac{\partial u}{\partial r}=0.$$ Thus $(1)$ becomes to $$ \frac{\partial^2 u}{\partial \theta ^2}=0 $$ and we have $u(z)=a\theta +b=a\arg z +b$.