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 $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ The function $f$ is then a function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$, called a function with separable polar variables. Show that, if $f$ is harmonic and not identically zero on $\mathbb{R}^2 \setminus \{(0,0)\}$, then there exists a real number $\lambda$ such that $u$ is a solution of the differential equation (II.1) $$r^2 z''(r) + r z'(r) - \lambda z(r) = 0$$ and $v$ is a solution of the differential equation (II.2) $$z''(\theta) + \lambda z(\theta) = 0$$
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
$$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$
The function $f$ is then a function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$, called a function with separable polar variables.
Show that, if $f$ is harmonic and not identically zero on $\mathbb{R}^2 \setminus \{(0,0)\}$, then there exists a real number $\lambda$ such that $u$ is a solution of the differential equation (II.1)
$$r^2 z''(r) + r z'(r) - \lambda z(r) = 0$$
and $v$ is a solution of the differential equation (II.2)
$$z''(\theta) + \lambda z(\theta) = 0$$