The elementary functions according to Liouville and Ritt are the functions of a complex variable built up by applying exponentiation, logarithms and/or algebraic operations finitely often. That means, the elementary functions are those functions built up by applying finitely often only $\exp$, $\ln$ and/or unary or multiary univalued algebraic functions.

J. F. Ritt proved in [Ritt 1925] that if a bijective elementary function $f$ has an elementary inverse, then $f=\psi_n\circ\ ...\ \circ \psi_1$, where each $\psi_i$ is either an algebraic function of one variable or else $\exp$ or $\ln$. That means, there exists a representation of $f$ as elementary function that doesn't contain multiary algebraic functions.

R. H. Risch proved Ritt's result in [Risch 1979], in the sharpened form above, from his structure theorem of elementary functions given there.

How can Ritt's theorem on elementary invertible bijective elementary functions be extended to further classes of functions?

Ritt's theorem is important for inverting functions in closed form and for solving equations in closed form. The theorem means, an elementary function for which only function expressions exist which also contain multiary algebraic functions doesn't have an elementary inverse.

Risch's proof shows when a composition $f^{-1}\circ f$ is algebraically dependent over the differential field of $f$. How can Risch's structure theorem on algebraic dependence of elementary functions and Ritt's theorem deduced from that be extended to further classes of functions?

There are a number of further classes of functions representable by a differential field, e.g. subfields of the elementary functions, the Liouvillian functions, subfields of Liouvillian fields, and functions that are generated by (generalized) composition of algebraic functions together with some (e.g. unary univalued) special functions. There are already a few structure theorems on algebraic dependence in such differential fields. See e.g. [Epstein, Caviness 1979] and [Singer/Saunders/Caviness 1985]. But how can these structure theorems be applied to prove theorems for inverses in these differential fields, like Ritt's theorem, as Risch did it?

My guess is that a bijective (generalized) composition $f$ of algebraic functions together with some differentiable unary univalued transcendental functions for which only function expressions exist that contain multiary univalued algebraic functions doesn't have an inverse in the differential field generated by the functions that build the function expression of $f$.

At least the defining equations of the inverse $f^{-1}$ of $f$ with $y=f^{-1}(z)$ $\ \ $ $f^{-1}(f(y))=z$ and $f(f^{-1}(y))=z$ cannot be solved for $y$ by only transforming the equations only by inverting the operations (functions) directly readable from this equations. That's because the inverse relation of a multiary univalued algebraic function is multivalued and therefore not a function and in particular not an allowed function. But it has to be proved if there are other functions $f^{-1}$ contained in the field generated by the functions that build the function expression of $f$.

A recent article is [Belov-Kanel/Malistov/Zaytsev 2020].
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[Ritt 1925] Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90

[Ritt 1948] Ritt, J. F.: Integration in finite terms. Liouville's theory of elementary methods. Columbia Univbersity Press, New York, 1948

[Rothstein, Caviness 1979] Rothstein, M.; Caviness, B. F.: A structure theorem for exponential and primitive functions. SIAM J. Comput. 8 (1979) 357-367

[Risch 1979] Risch, R. H.: Algebraic Properties of the Elementary Functions of Analysis. Amer. J. Math 101 (1979) (4) 743-759

[Epstein, Caviness 1979] Epstein, H. I.; Caviness, B. F.: A structure theorem for the elementary functions and its application to the identity problem. Int. J. Comp. Inf. Sci. 8 (1979) (1) 9-37

[Singer/Saunders/Caviness 1985] Singer, M. F.; Saunders, B. D.; Caviness, B. F.: An extension of Liouville's theorem on integration in finite terms. SIAM J. Comput. 14 (1985) (4) 966-990

[Belov-Kanel/Malistov/Zaytsev 2020] Belov-Kanel, A.; Malistov, A.; Zaytsev, R.: Solvability of equations in elementary functions. Journal of Knot Theory and Its Ramifications 29 (2020) (2) 204-205


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