Since $\displaystyle u = \frac{1}{2}\log(x^2+y^2)$ is, as already answered in this forum, harmonic for $\displaystyle x,y > 0$, and $\Delta u = 0,$ i don't get why this function is not harmonic in $\displaystyle x=y=0$. Is there a condition for harmonic functions not to have singularities?
2026-02-22 19:52:07.1771789927
Why is $ u=\log(\sqrt{x^2+y^2})$ not harmonic for $x^2 + y^2 <1$?
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By definition harmonic functions are twice continuously differentiable everywhere on their domain. In particular that implies they are continuous so they can't have singularities.