In this subsection II.C, we consider two functions of class $\mathcal{C}^2$, $u : \mathbb{R}^{*+} \rightarrow \mathbb{R}$ and $v : \mathbb{R} \rightarrow \mathbb{R}$ and we set $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ We now assume $\lambda \neq 0$. Solve (II.1) on $\mathbb{R}^{+*}$: $$r^2 z''(r) + r z'(r) - \lambda z(r) = 0$$ One may consider, justifying its existence, a function $Z$ of class $\mathcal{C}^2$ on $\mathbb{R}$ such that, for all $r > 0$, $z(r) = Z(\ln(r))$.
In this subsection II.C, we consider two functions of class $\mathcal{C}^2$, $u : \mathbb{R}^{*+} \rightarrow \mathbb{R}$ and $v : \mathbb{R} \rightarrow \mathbb{R}$ and we set
$$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$
We now assume $\lambda \neq 0$. Solve (II.1) on $\mathbb{R}^{+*}$:
$$r^2 z''(r) + r z'(r) - \lambda z(r) = 0$$
One may consider, justifying its existence, a function $Z$ of class $\mathcal{C}^2$ on $\mathbb{R}$ such that, for all $r > 0$, $z(r) = Z(\ln(r))$.