We seek to determine the non-zero harmonic functions on $\mathbb{R}^2$ with separable variables, that is, functions $f$ that can be written in the form $f(x,y) = u(x)v(y)$. We are given two functions $u$ and $v$, of class $\mathcal{C}^2$ on $\mathbb{R}$, not identically zero, and we set $$\forall (x,y) \in \mathbb{R}^2, \quad f(x,y) = u(x)v(y)$$ We assume that $f$ is harmonic on $\mathbb{R}^2$. Show that there exists a real constant $\lambda$ such that $u$ and $v$ are solutions respectively of the equations $$z'' + \lambda z = 0 \quad \text{and} \quad z'' - \lambda z = 0$$
We seek to determine the non-zero harmonic functions on $\mathbb{R}^2$ with separable variables, that is, functions $f$ that can be written in the form $f(x,y) = u(x)v(y)$. We are given two functions $u$ and $v$, of class $\mathcal{C}^2$ on $\mathbb{R}$, not identically zero, and we set
$$\forall (x,y) \in \mathbb{R}^2, \quad f(x,y) = u(x)v(y)$$
We assume that $f$ is harmonic on $\mathbb{R}^2$.
Show that there exists a real constant $\lambda$ such that $u$ and $v$ are solutions respectively of the equations
$$z'' + \lambda z = 0 \quad \text{and} \quad z'' - \lambda z = 0$$