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 $\mathbb{R}^2$. Give, depending on the sign of $\lambda$, the form of harmonic functions with separable variables.
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 $\mathbb{R}^2$.
Give, depending on the sign of $\lambda$, the form of harmonic functions with separable variables.