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.

Answer

Liouville’s theorem deals with an elementary differential extension, composition is considered there. 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.

Attribution
Source : Link , Question Author : RghtHndSd , Answer Author : IV_

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