Timeline for How to solve $f(f(x)) = \cos(x)$?
Current License: CC BY-SA 4.0
30 events
| when toggle format | what | by | license | comment | |
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| Oct 26, 2021 at 19:43 | comment | added | Cosmas Zachos | Might appreciate this. | |
| Jul 18, 2020 at 19:32 | history | edited | Rodrigo de Azevedo | CC BY-SA 4.0 |
Minor improvements.
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| Apr 28, 2020 at 13:52 | answer | added | Lasse Rempe | timeline score: 8 | |
| Mar 15, 2017 at 13:32 | answer | added | Gottfried Helms | timeline score: 4 | |
| Oct 18, 2016 at 19:36 | answer | added | user78249 | timeline score: 3 | |
| S Feb 11, 2016 at 13:13 | history | suggested | CommunityBot |
Added tag.
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| Feb 11, 2016 at 12:53 | review | Suggested edits | |||
| S Feb 11, 2016 at 13:13 | |||||
| Feb 11, 2016 at 12:39 | history | edited | Salvo Tringali | CC BY-SA 3.0 |
Added a backslash in the title
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| Feb 11, 2016 at 12:35 | history | edited | Denis Serre | CC BY-SA 3.0 |
edited title
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| May 31, 2012 at 12:15 | comment | added | Joel David Hamkins | Anixx, the method of my solution shows that if one augments the domain and range to include even an additional tiny interval, let alone the entire complex plane, then it is easy to find numerous half-iterates that work on the original domain: simply map the old domain into the new augmented part, and map the new augmented part to the desired value. | |
| May 31, 2012 at 11:36 | comment | added | Anixx | To clarify what solution I found: I found a complex-valued function that is half-iterate of real-valued cosine. That is repeated twice on any real argument it gives cosine. It is not a half-iterate of cosine on the whole complex plane, only on reals. | |
| May 31, 2012 at 10:44 | comment | added | Anixx | I modifyed and re-posted my answer to avoid it becoming a community wiki. I improved the calculation and plots quality and also found a function that works as a cosine half-iterate on the whole real axis. | |
| May 30, 2012 at 2:47 | comment | added | Anixx | @Gerald Edgar, does your solution coincide with that of mine (see below mathoverflow.net/questions/17605/how-to-solve-ffx-cosx/… )? I wonder why it has been downvoted... | |
| May 29, 2012 at 11:57 | comment | added | Gerald Edgar | @Noah: for complex solution, see my reply mathoverflow.net/questions/17605/how-to-solve-ffx-cosx/… | |
| May 29, 2012 at 3:48 | comment | added | Noah Snyder | Sergei's impossibility argument only applies to real functions, right? (Because it uses monotonicity.) Cosine is a perfectly healthy complex function, so you could ask the same question in the complex setting. As far as I can tell no one has answered that version of the question conclusively yet. | |
| May 29, 2012 at 3:09 | answer | added | Anixx | timeline score: 10 | |
| Jul 30, 2011 at 16:52 | history | edited | Michael Hardy | CC BY-SA 3.0 |
The backslash not only prevents italicization, but in some contexts results in proper spacing.
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| Nov 3, 2010 at 20:58 | answer | added | Anixx | timeline score: 89 | |
| Mar 11, 2010 at 13:19 | answer | added | user4503 | timeline score: 6 | |
| Mar 10, 2010 at 1:22 | comment | added | Anonymous | After Sergey Ivanov's solution, I wonder how many discontinuity points a solution of $f(f(x)) = \cos(x)$ must have? | |
| Mar 9, 2010 at 23:15 | answer | added | user4503 | timeline score: 7 | |
| Mar 9, 2010 at 23:02 | comment | added | Gerry Myerson | There is a collection of references and links for the general problem of solving $f(f(x))=g(x)$ for $f$, given $g$, at reglos.de/lars/ffx.html | |
| Mar 9, 2010 at 22:44 | comment | added | user4503 | I would have liked to see a continuous solution, however it seems like you already showed, that such a solution cannot exist. | |
| Mar 9, 2010 at 18:55 | answer | added | Sergei Ivanov | timeline score: 69 | |
| Mar 9, 2010 at 17:49 | comment | added | Sergei Ivanov | Are you looking for continuous solutions or arbitrary maps? | |
| Mar 9, 2010 at 15:10 | answer | added | Joel David Hamkins | timeline score: 29 | |
| Mar 9, 2010 at 15:01 | answer | added | Dmytro Yeroshkin | timeline score: 5 | |
| Mar 9, 2010 at 14:50 | answer | added | HenrikRüping | timeline score: 5 | |
| Mar 9, 2010 at 14:47 | answer | added | Gerald Edgar | timeline score: 13 | |
| Mar 9, 2010 at 14:30 | history | asked | user4503 | CC BY-SA 2.5 |