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 $\lambda \neq 0$. Give a necessary and sufficient condition for (II.2)
$$z''(\theta) + \lambda z(\theta) = 0$$
to admit non-zero $2\pi$-periodic solutions. Give these solutions.