harmonic function. How to prove?

Question

I've with prove if $1 \over |x|$ is a harmonic function. I know with for a harmonic function, $f_{xx}+f_{yy}=0$, but I don't know how to derivate ${1 \over |x|} dx$. And I don't know how to derivate $f_{yy}$ for this function because it haven't $y$ terms. Can you help me to solve this problem? Thanks!

Answer 1

This $|x|$ should mean $\sqrt{x^2+y^2+z^2}$. This would be harmonic on $\mathbb{R}^3\backslash \{0\}$.

Answer 2

Harmonic function $\phi$ satisfies Laplace equation: $\Delta \phi = 0$ Switching to spherical coordinates give: $$\frac{1}{r^2}\frac{\delta}{\delta r}r^2\frac{\delta}{\delta r} * \frac{1}{r} = 0$$ It is not harmonic in any other dimension except $n = 3$