Determine the radial harmonic functions of $\mathbb{R}^2 \setminus \{(0,0)\}$, that is, the functions $f$ belonging to $\mathcal{H}\left(\mathbb{R}^2 \setminus \{(0,0)\}\right)$ such that $(r,\theta) \mapsto f(r\cos(\theta), r\sin(\theta))$ is independent of $\theta$.