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 not identically zero, then $v$ is $2\pi$-periodic.
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 not identically zero, then $v$ is $2\pi$-periodic.