Let $f$ be a real function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$. We set, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$,
$$g(r,\theta) = f(r\cos(\theta), r\sin(\theta))$$
Determine the radial harmonic functions of $\mathbb{R}^2 \setminus \{(0,0)\}$, that is, the functions $f$ belonging to $\mathcal{H}(\mathbb{R}^2 \setminus \{(0,0)\})$ such that $(r,\theta) \mapsto f(r\cos(\theta), r\sin(\theta))$ is independent of $\theta$.