Examples for 2-dimensional real valued harmonic functions

Question

Given: $$f: \mathbb{R}^2 \to \mathbb{R},\space \Delta{f}=0, \space\frac{\partial^2{f}}{\partial{x}^2_i} \neq 0, i \in\{1,2 \}.$$ Are there examples of such functions?

Answer 1

Just to prevent this question reappearing in the unsolved list:

The simplest example I can think of would be $f(x,y) = x^2 - y^2$.

That this is harmonic can be seen by either calculating $\Delta{f}$ and finding that is identically zero, or simply by noticing that this function is the real part of the holomorphic function $f(z) = z^2$.

When you state the condition $\frac{\partial^2{f}}{\partial{x}^2_i} \neq 0$, it might not be 100% clear whether you meant "is not identically zero" or "is never zero", but this particular example does both, since $\frac{\partial^2{f}}{\partial{x}^2} = 2$ everywhere.