Timeline for generalisation of Cauchy-Riemann equations to 3D
Current License: CC BY-SA 3.0
7 events
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| May 23, 2018 at 22:29 | comment | added | Ali Taghavi | @Denis In the following post I considered another kind of "preserving", that is invariant under derivation operator(instead of composition).Do you have some extra ideas,(aside of existing answer) on this question?Can I ask you to give some comment on this question? mathoverflow.net/questions/162598/… | |
| May 15, 2012 at 8:50 | history | edited | Denis Serre | CC BY-SA 3.0 |
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| May 15, 2012 at 5:11 | history | edited | Denis Serre | CC BY-SA 3.0 |
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| May 15, 2012 at 1:58 | comment | added | curious | ah yes my bad. Thank you for straighting me out, my example was indeed nonsense. Let me check if inversion preserves harmonicity... any other examples? or is it an exhaustive list? | |
| May 15, 2012 at 1:03 | comment | added | Thomas Klimpel | The comment above contains frustratingly many typos (wrong parenthesis in the expression for $\Delta (u \circ \psi)$, $x\mapsto 1/||x||^n$ instead of $x\mapsto x/||x||^n$), but the message should be clear nevertheless. | |
| May 15, 2012 at 0:58 | comment | added | Thomas Klimpel | @curious Note that $\Delta(u\circ \psi)=|f'|^2((\partial_1^2+\partial_2^2)u+\partial_3^2 u)\circ \psi$, which is not proportional to $(\Delta u)\circ \psi$ if $|f'|^2\neq 1$ and $\partial_3^2u\neq 0$. So Denis Serre is right that your "examples" usually don't work. However, I looked up the theorem due to Liouville in my copy of Marcel Berger's "Géométrie 1", and certain inversions also seem to preserve harmonicity. I guess they are $x \mapsto 1/||x||^n$, but I would have to check. Of course, these inversions are not defined everywhere in $\mathbb R^3$, so the remark is only misleading. | |
| May 14, 2012 at 16:27 | history | answered | Denis Serre | CC BY-SA 3.0 |