As proposed in this answer, I wonder if the answer to following question is known.

Let $E = E_0$ be the set of elementary functions. For each $i > 0$, inductively define $E_i$ to be the closure of the set of functions whose derivative lies in $E_{i-1}$ with respect to multiplication, inversion, and composition. Does there exist an integer $n$ such that $E_n = E_{n+1}$?

This seems like such a natural generalization of Liouville's theorem, it has to have been asked before. After a couple of quick internet searches, I can't seem to find anything.

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    $\begingroup$ I feel like you need to throw some kind of closure operation in there -- something like "Define $E_i$ to be the set of all functions that can be formed by a composition of functions whose derivatives lie in $E_{i-1}$." But maybe that is unnecessary. $\endgroup$ – mweiss Jun 17 '14 at 0:01
  • $\begingroup$ Indeed, it is necessary. Thanks. $\endgroup$ – RghtHndSd Jun 17 '14 at 0:07
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    $\begingroup$ Similar question has been asked before math.stackexchange.com/questions/686445/… but alas no answers. $\endgroup$ – Conifold Jun 17 '14 at 1:03
  • $\begingroup$ And how would this relate to notions of computability (e.g Turing Machine), since an elementary functin can be a symbol along with the other (finite) symbols of "addition", "nth power", "division" etc.. This may indeed be related.. And provide sth analogous to an "algebraic closure", but if taken in a computable sense, this would be non-decidable.. $\endgroup$ – Nikos M. Jun 17 '14 at 20:36
  • $\begingroup$ @NikosM.: I don't understand how "notions of computability" can be related to my question. What does this mean? Perhaps you are asking "do the E_i's consist of computable functions"? Also regarding the last sentence, precise what is it that "would be non-decidable"? $\endgroup$ – RghtHndSd Jun 17 '14 at 21:04

Liouville's theorem deals with an elementary differential extension. Composition is considered because algebraic operations are allowed, and the identity function is algebraic. But your problem contains the additional operation inversion.

Therefore your problem is not a generalization of Liouville's theorem but a different task.

$E_{i+1}\setminus E_{i}$ contains the non-elementary antiderivatives of the functions from $E_{i}$ and the non-elementary inverses of the functions from $E_{i}$.

With a generalization of the theorem of Ritt of Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90 which I hope to prove, one could show there are elementary functions in each of your $E_{i}$ that have a non-elementary inverse.

Your $E_{i}$ are therefore no differential fields and you cannot apply Liouville's theorem. Therefore your problem cannot be solved by the Liouville theory treated in the literature.


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