How do you show that every non-constant real valued harmonic function on $\mathbb{R}^n$ has a zero? What about $\mathbb{R}^2$ \ $\{0\}$?
2026-03-25 09:27:08.1774430828
How do you show that every non-constant real valued harmonic function on $\mathbb{R}^n$ has a zero? What about $\mathbb{R}^2$ \ $\{0\}$?
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If it is non-constant, then it can't be either bounded from above or from below. You can use the short proof from here. In particular, it must have a zero.
On $\mathbb{R}^2\setminus\{0\}$ you can take $1/\sqrt{x^2+y^2}$.