# Generalization of Liouville's theorem

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.

• 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. – mweiss Jun 17 '14 at 0:01
• Indeed, it is necessary. Thanks. – RghtHndSd Jun 17 '14 at 0:07
• Similar question has been asked before math.stackexchange.com/questions/686445/… but alas no answers. – Conifold Jun 17 '14 at 1:03
• 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.. – Nikos M. Jun 17 '14 at 20:36
• @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"? – RghtHndSd Jun 17 '14 at 21:04

$E_{i+1}\setminus E_{i}$ 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.