Timeline for Topological degree of differentiable map using line integrals?
Current License: CC BY-SA 4.0
27 events
| when toggle format | what | by | license | comment | |
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| S Nov 4, 2023 at 11:58 | history | bounty ended | António Borges Santos | ||
| S Nov 4, 2023 at 11:58 | history | notice removed | António Borges Santos | ||
| Nov 2, 2023 at 16:21 | vote | accept | António Borges Santos | ||
| Nov 2, 2023 at 12:26 | comment | added | António Borges Santos | @CarloBeenakker I just opened a bounty, since I think I still don't quite know why this expression is an integer. | |
| S Nov 2, 2023 at 12:25 | history | bounty started | António Borges Santos | ||
| S Nov 2, 2023 at 12:25 | history | notice added | António Borges Santos | Authoritative reference needed | |
| Nov 1, 2023 at 8:12 | history | edited | António Borges Santos | CC BY-SA 4.0 |
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| Oct 31, 2023 at 20:37 | comment | added | António Borges Santos | @M.G. that's interesting, do you think you can make the connection explicit? | |
| Oct 31, 2023 at 20:11 | comment | added | M.G. | @AntónioBorgesSantos: have you looked into topological degree of differentiable maps? | |
| Oct 31, 2023 at 19:59 | comment | added | António Borges Santos | thank you very much, that makes sense... | |
| Oct 31, 2023 at 19:56 | comment | added | Carlo Beenakker | I should not have used the word multiplicity, it refers to real functions; in the complex plane winding number is the correct word; a function $f(z)=z^p$ has winding number $p$ in the complex plane, and multiplicity $p$ of its root on the real axis, but for your function $x+iy^2$ only winding number makes sense (you could argue it has multiplicity 1 along the real axis and 2 along the imaginary axis, so that is not a unique definition) | |
| Oct 31, 2023 at 19:48 | comment | added | António Borges Santos | @CarloBeenakker thank you for your answer, but then the integral is not $2\pi$ times the multiplicity of the root? | |
| Oct 31, 2023 at 19:45 | comment | added | Carlo Beenakker | these two function give different values, one has winding number 1, the other winding number 0; I have worked this out in the answer box. | |
| Oct 31, 2023 at 19:44 | answer | added | Carlo Beenakker | timeline score: 3 | |
| Oct 31, 2023 at 19:40 | history | edited | António Borges Santos | CC BY-SA 4.0 |
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| Oct 31, 2023 at 17:30 | history | edited | Michael Hardy | CC BY-SA 4.0 |
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| Oct 31, 2023 at 16:17 | comment | added | António Borges Santos | @CarloBeenakker I see, but then $f(x,y)=x+iy$ and $f(x,y)=x+iy^2$ give the same value? I am still not quite sure I see what computation implies this invariance... | |
| Oct 31, 2023 at 16:10 | comment | added | Carlo Beenakker | You need isolated roots, differentiable is not enough. For example, this example you mentioned earlier of $f(x)=x_1$ fails, it vanishes on the entire imaginary axis. | |
| Oct 31, 2023 at 16:09 | history | edited | António Borges Santos | CC BY-SA 4.0 |
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| Oct 31, 2023 at 15:48 | comment | added | António Borges Santos | @CarloBeenakker but here we are not considering holomorphic functions but just real differentiable functions? | |
| Oct 31, 2023 at 15:11 | comment | added | Carlo Beenakker | see en.wikipedia.org/wiki/Argument_principle | |
| Oct 31, 2023 at 13:32 | comment | added | António Borges Santos | @CarloBeenakker does this winding number have a name or do you have a reference where these properties are discussed? | |
| Oct 31, 2023 at 12:33 | history | edited | António Borges Santos | CC BY-SA 4.0 |
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| Oct 31, 2023 at 12:32 | history | undeleted | António Borges Santos | ||
| Oct 31, 2023 at 10:32 | history | deleted | António Borges Santos | via Vote | |
| Oct 31, 2023 at 10:04 | comment | added | Carlo Beenakker | it's independent of $\epsilon$, the integral is $2\pi$ times the multiplicity of the root. | |
| Oct 31, 2023 at 9:53 | history | asked | António Borges Santos | CC BY-SA 4.0 |