Timeline for How complicated can an elementary antiderivative get?
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| S Feb 5 at 3:04 | history | bounty ended | CommunityBot | ||
| S Feb 5 at 3:04 | history | notice removed | CommunityBot | ||
| Feb 4 at 2:59 | history | edited | pie | CC BY-SA 4.0 |
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| Feb 3 at 22:55 | comment | added | mick | I see no reason to assume long solutions for short elementary integrands with low coefficients. $\int dx/(2 + sin^5) $ is short for example. Afterall if the number of compositions and products are small, we probably do not need to use many tricks. Especially if we allow hypergeo and other special functions and sums over zero's of a given function. Show me a counterexample !? | |
| Feb 2 at 1:34 | comment | added | Timothy Chow | @WillSawin Hmmm...good point! That does seem to be correct. I still don't expect that anything like Ackermann is possible, though. | |
| Feb 1 at 17:26 | comment | added | Will Sawin | @TimothyChow Won't you get faster growth just because the definition of elementary function involves composition so we can write the integral of $\frac{x^n-1}{x-1}$ where $n$ is a tower of exponentials of length $m$ in $O(m)$ characters? But it's probably not possible to express the integral with fewer than $n$ characters. | |
| S Jan 28 at 1:22 | history | bounty started | pie | ||
| S Jan 28 at 1:22 | history | notice added | pie | Draw attention | |
| Jan 23 at 19:30 | comment | added | LSpice | Your link went to a comment on your MSE question, whereas it seemed that you meant to link to the question itself. I edited accordingly. | |
| Jan 23 at 19:29 | history | edited | LSpice | CC BY-SA 4.0 |
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| Jan 23 at 19:22 | comment | added | pie | @AndyPutman Ok I will not edit this again. I am sorry for that. | |
| Jan 23 at 19:20 | comment | added | Andy Putman | @pie: I think it's best to not make so many edits. Each edit pushes the question to the top of the queue again, so it is unfair to other people who are trying to get attention for their questions. | |
| Jan 23 at 19:15 | comment | added | pie | @AndyPutman most of them are grammar edits I am not good t english and so I am not very sure want to use to write this (for example do I use "how much complicated" or "how complicated") , you will notice many of my question on MSE or HSM have very bad grammar or spelling, If it is annoying I will not edit this question again I apologise for this | |
| Jan 23 at 19:11 | comment | added | Andy Putman | I think that at this point it would be best if you stopped the frequent minor edits. If you can't get the question right after 11 edits, then I would stop and reflect about what exactly you're trying to get from it. | |
| Jan 23 at 18:58 | history | edited | pie | CC BY-SA 4.0 |
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| Jan 23 at 16:11 | comment | added | Max Lonysa Muller | I think it is also important in this regard to consider how one measures the complexity of the resulting antiderivatives. If one employs the Kolmogorov complexity as a measure, on might be interested in the following article: link.springer.com/chapter/10.1007/978-3-642-21875-0_9 | |
| Jan 23 at 12:29 | history | edited | pie | CC BY-SA 4.0 |
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| Jan 23 at 9:32 | history | edited | pie | CC BY-SA 4.0 |
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| Jan 23 at 0:19 | history | edited | pie |
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| Jan 22 at 21:32 | history | edited | pie | CC BY-SA 4.0 |
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| Jan 22 at 18:42 | history | edited | Sam Hopkins | CC BY-SA 4.0 |
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| Jan 22 at 18:10 | history | edited | pie | CC BY-SA 4.0 |
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| Jan 22 at 17:57 | history | edited | pie | CC BY-SA 4.0 |
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| Jan 22 at 14:01 | comment | added | Timothy Chow | I agree with Command Master's comment. See this MO question, and in particular the paper by Daniel Schultz, for some indication of how complicated an elementary antiderivative can get. Offhand, I don't know an upper bound, but my impression is that the most computationally expensive part is some kind of Gröbner basis calculation. So I'd guess that there's a doubly exponential upper bound. Certainly nothing like TREE or even Ackermann. | |
| Jan 22 at 10:41 | history | edited | pie | CC BY-SA 4.0 |
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| Jan 22 at 10:39 | comment | added | Daniel Weber | It might be possible to extract an upper bound for the possible length from the Risch algorithm | |
| Jan 22 at 10:29 | comment | added | mlk | Something like $\int \sin^n dx$ for large n should result in lots and lots of terms when calculating it in the standard way of repeated partial integrations, but I have no idea how to prove that there is no nicer closed form. | |
| Jan 22 at 10:09 | review | Close votes | |||
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| Jan 22 at 9:58 | history | edited | pie | CC BY-SA 4.0 |
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| Jan 22 at 9:52 | history | asked | pie | CC BY-SA 4.0 |