Harmonic harmonic

Question

Determine the form of functions $Y:R \to R$ such that the function in $(x,z) \to Y(x/z)$ is harmonic .

Can I find this form of function ? I need find function Y such that Laplace $Y = 0$ ?

Answer 1

Let $u(x,z)=Y(x/z)$. First of all $Y$ should be a $C^2-$ function.

Now compute $u_{xx}$ and $u_{zz}$.

$u$ is harmonic $ \iff u_{xx}+u_{zz}=0$.

It is your turn to show that this is equivalent to $Y''=0$.