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?
Asked
Active
Viewed 958 times
1
1 Answers
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.
Old John
- 19,569
- 3
- 59
- 113
and yes, Old John's example is Harmonic since the definition only requires that $\Delta f=0$
– b00n heT Jan 12 '14 at 11:06