Let $f$ be a real function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$. We set, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, $$g(r,\theta) = f(r\cos(\theta), r\sin(\theta))$$ Show that $f$ belongs to $\mathcal{H}\left(\mathbb{R}^2 \setminus \{(0,0)\}\right)$ if and only if, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, $$r^2 \frac{\partial^2 g}{\partial r^2}(r,\theta) + \frac{\partial^2 g}{\partial \theta^2}(r,\theta) + r\frac{\partial g}{\partial r}(r,\theta) = 0$$
Let $f$ be a real function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$. We set, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$,
$$g(r,\theta) = f(r\cos(\theta), r\sin(\theta))$$
Show that $f$ belongs to $\mathcal{H}\left(\mathbb{R}^2 \setminus \{(0,0)\}\right)$ if and only if, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$,
$$r^2 \frac{\partial^2 g}{\partial r^2}(r,\theta) + \frac{\partial^2 g}{\partial \theta^2}(r,\theta) + r\frac{\partial g}{\partial r}(r,\theta) = 0$$