Harmonic function

Consider the points $O(0, 0)$ and $A(1, 0)$ . Let $\Gamma(x, y)$ be a point of the plane such that $y>0$ . Set $\varphi(x, y)$ to be the angle that is defined by $O\Gamma$ and $A \Gamma$ . ( the one that is less than $\pi$ .) Prove that the function $\varphi(x, y)$ is harmonic.

Solution [by Demetres Skouteris]

The complex function $\log \left( 1 - \frac{1}{z} \right)$ has a holomorphic branch in the half plane $\mathfrak{Im}(z)>0$ and its imaginary part is the desired angle. Hence, the function of the angle is harmonic.