When can an invertible function be inverted in closed form?

Question

The Risch algorithm answers the question: "When can a function be integrated in closed form?", see: https://en.wikipedia.org/wiki/Symbolic_integration Is anyone aware of any work that answers the related question, "When can an invertible function be inverted in closed form?" By closed form, I wish to exclude infinite series expansions, unless they describe a special function. I would be happy just to see a short list of some explicitly invertible functions.

Answer 1

I recommend the following paper: MR1501299 Ritt, J. F. Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925), no. 1, 68–90. (freely available on the web). It indeed gives a short list. For more recent results there is a book A. Khovanski, Topological Galois theory.

Of course you should specify more exactly what do you mean by a closed form. In Ritt (and other papers on the subject), algebraic functions are considered "elementary". If from your point of view they are not "closed forms", you may look to another paper by Ritt:

MR1501211 Ritt, J. F. On algebraic functions which can be expressed in terms of radicals. Trans. Amer. Math. Soc. 24 (1922), no. 1, 21–30.

MR1501229 Ritt, J. F. Errata: "On algebraic functions which can be expressed in terms of radicals'' Trans. Amer. Math. Soc. 24 (1922), no. 4, 324.

http://www.ams.org/journals/tran/1925-027-01/S0002-9947-1925-1501299-9/S0002-9947-1925-1501299-9.pdf

On the other hand, many people consider Lambert's function (the inverse of $xe^x$) "closed form". It is certainly not elementary.

Answer 2

Closed-form functions need the definition which set of functions is allowed to represent the function. Take e.g. the algebraic definition of the Elementary functions by Liouville and Ritt (Wikipedia: Elementary function).

There are only few publications that prove that the function given there doesn't have an inverse in closed form. Examples are the proof that the solutions of Kepler's equation are not elementary functions* and the proof that LambertW is not an elementary function**.
*)
Joseph Liouville
Ritt, J. F.: Integration in finite terms. Liouville's theory of elementary methods. Columbia University Press, New York, 1948
**)
Why is it that the Lambert W relation cannot be expressed in terms of elementary functions?
Dubinov, A.; Galidakis, Y.: Explicit solution of the Kepler equation. Physics of Particles and Nuclei Letters 4 (2007) 213-216
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1.) Ritt and Risch

Ritt's work Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90 treats the topic in general. But it deals only with the elementary functions, and Ritt's method of proof seems to be unfortunately only for the elementary functions.

Ritt's theorem is proved also by Risch in [Risch 1979] Risch, R. H.: Algebraic Properties of the Elementary Functions of Analysis. Amer. J. Math 101 (1979) (4) 743-759. I assume it is possible to extend this proof and Ritt's theorem to other classes of functions. See my question How to extend Ritt's theorem on elementary invertible bijective elementary functions.

Another reference is the last theorem in Rosenlicht, M.: On the explicit solvability of certain transcendental equations. Publications mathématiques de l'IHÉS 36 (1969) 15-22. Rosenlicht writes: "The preceding theorem is a powerful tool for finding elementary solutions, if such exist, of certain types of transcendental equations, or for proving their nonexistence." Unfortunately, this method is applicable only for certain kinds of equations. But the method is applicable also for some other classes of functions which can be represented by a field.

How classes of functions can be represented as a set of functions generated by a field, is treated in differential algebra and is described e.g. in section 1 of Davenport, J. H.: What Might "Understand a Function" Mean. In: Kauers, M.; Kerber, M., Miner, R.; Windsteiger, W.: Towards Mechanized Mathematical Assistants. Springer, Berlin/Heidelberg, 2007, page 55-65.
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It is possible to extend at least one part of Ritt's results directly as follows:

a)

Let
$K_0$ a field,
$n\in\mathbb{N}_+$,
$X,Y_1,...,Y_n$ sets.
$A\colon Y_1\times \cdots\times Y_n \mapsto A(Y_1\times ...\times Y_n), (x_1,...,x_n)\mapsto A(x_1,...,x_n)$ be a function, algebraic over the field $K_0$.
$f_1\colon X\to Y_1, x\mapsto f_1(x)$; ...; $f_n\colon X\to Y_n, x\mapsto f_n(x)$ be bijective transcendental functions.
$F\colon X\to Y, x\mapsto A(f_1(x),...,f_n(x))$ be a bijective function.
$K$ be the extension field of $K_0$ which is generated from $K_0$ only by applying algebraic operations, the identity function and the inverses of $f_1,...,f_n$.
If $f_1,...,f_n$ are pairwise algebraically dependent over $K_0$, then $F$ has an inverse in $K$.

That means, roughly spoken, each bijective iterated composition of unary univalued functions $f_i$ (maybe this can be extended to $n_i$-ary $n_i$-valued functions) is invertible in the algebraic closure of the differential field which is generated by the $f_i$ and their inverses.

This, if formulated as theorem, is important, because each function on an open domain can be made bijective by restriction of its domain. This plays a role at inverting functions and solving equations by partial inverses.

I gave an example in the answer to Algebraic solution to natural logarithm equations like $1-x+xln(-x)=0$.
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The following is an unproven assumption of mine:
If $trdeg_{K_0}K_0(f_1,f_2,...,f_n)>0$, then $F$ cannot have an inverse in $K$.

An indication for this assumption: The defining equations for the inverse of $F$, $F^{-1}(F(x))=x$ and $F(F^{-1}(x))=x$, cannot be solved by applying only functions from $K$. That means, it is not possible to calculate the inverse of $F$ by solving this equations on that way.

But Ritt's proof goes further: Ritt $proves$ that a corresponding elementary function $F$ cannot have an inverse in the elementary functions.

b)

One can easily prove the

Theorem:
Let $n \in \mathbb{N}_0$,
$f_{1},...,f_{n}$ bijective functions,
$f=f_{n}\circ f_{n-1}\circ\ ...\ \circ f_{2}\circ f_{1}$ a bijective function,
$\phi$ the inverse of $f$.
Then $\phi=\phi_{1}\ \circ\ \phi_{2}\ \circ\ ...\ \circ\ \phi_{n-1}\ \circ\ \phi_{n}$, wherein for all $i$ with $1\leq i\leq n$, $\phi_{i}$ is the inverse of $f_{i}$.

For inverses in the elementary functions and LambertW, I could apply this theorem for my answer at https://math.stackexchange.com/questions/2309691/equations-solvable-by-lambert-function/2527410#2527410.
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2.) Lin and Chow

As I found by an obvious but possibly new theorem (Proof Check: Nonexistence of the inverse function in a given class of functions), there is a connection between solvability of particular kinds of equations of the form $f(z)=\alpha$ in a given set of numbers and invertibility of functions $f$ in a given set of functions.

Nonexistence theorems for solutions of irreducible polynomial equations $P(z,e^z)=0$ in the Elementary numbers and in the Explicit elementary numbers, respectively are given in:
Ferng-Ching Lin: Schanuel's Conjecture Implies Ritt's Conjectures. Chin. J. Math. 11 (1983) (1) 41-50
Chow, T.: What is a closed-form number. Am. Math. Monthly 106 (1999) (5) 440-448

Are there further nonexistence statements known for solutions of equations in other classes of numbers?
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3.) Arnold and Khovanskii

The topological method of Vladimir Arnold and Askold Khovanskii allows the extension of the problem to a very large class of functions.

Khovanskii, A.: Topological Galois Theory - Solvability and Unsolvability of Equations in Finite Terms. Springer 2014,
Khovanskii, A.: One dimensional topological Galois theory. 2019:

"Definition. A multivalued analytic function of one complex variable is called a $S$-function, if the set of its singular points is at most countable."

"Theorem 3 (on stability of the class of $S$-functions). The class $S$ of all $S$-functions is stable under the following operations:

1. differentiation, i. e. if $f∈S$, then $f′ ∈ S$;
2. integration, i. e. if $f ∈ S$ and $g′ = f$, then $g ∈ S$;
3. composition, i. e. if $g$, $f ∈ S$, then $g ◦ f ∈ S$;
4. meromorphic operations, i. e. if $f_i ∈ S$, $i=1,...,n$, the function $F(x_1,...,x_n)$ is a meromorphic function of $n$ variables, and $f=F(f_1,...,f_n)$, then $f ∈ S$;
5. solving algebraic equations, i. e. if $f_i ∈ S$, $i=1,...,n$, and $f^n+f_1f^{n−1}+···+f_n=0$, then $f ∈ S$;
6. solving linear differential equations, i.e. if $f_i ∈ S$, $i=1,...,n$, and $f^{(n)}+f_1f^{(n−1)}+···+f_nf=0$, then $f ∈ S$.

Remark. Arithmetic operations and the exponentiation are examples of meromorphic operations, hence the class of $S$-functions is stable under the arithmetic operations and the exponentiation.

Corollary 4 (see [2]). If a multivalued function $f$ can be obtained from single valued $S$-functions by integration, differentiation, meromorphic operations, compositions, solutions of algebraic equations and linear differential equations, then the function $f$ has at most countable number of singular points.

Corollary 5. A function having uncountably many singular points cannot be represent by generalized quadratures. In particular it cannot be a generalized elementary function and it cannot be represented by k-quadratures or by quadratures."

"Theorem 6 (see [2]). The class of all S-functions, having a solvable monodromy group, is stable under composition, meromorphic operations, integration and differentiation."

Kanel-Belov, A.; Malistov, A.; Zaytsev, R.: Solvability of equations in elementary functions. 2019:
"Suppose we are trying to solve an equation $f(x) = a$ in the complex plane. If $f(x)$ is holomorphic and non-constant, then, by the well-known uniquenuess principle, for any $a$, the set of roots is discrete.
...
Corollary 2.2. For any N, the group induced by elementary functions and their compositions of depth not exceeding N is solvable
...
The last corollary means that if we can choose some curves, such that the group generated by their induced permutations is unsolvable, we will have proved the unsolvability in elementary functions.

Remark. Calculating the group induced by a given set of curves is rather difficult. This could be automated and done by a computer. Also, if the number of critical points is finite, there is only a finite number of curves, and therefore the whole proof could be done by a machine."

Zelenko, L.: Generic monodromy group of Riemann surfaces for inverses to entire functions of finite order. 2021:
"We consider the vector space $E_{\rho,p}$ of entire functions of finite order $\rho\in\mathbb{N}$, whose types are not more than $p > 0$, endowed with Frechet topology, which is generated by a sequence of weighted norms. We call a function $f\in E_{\rho,p}$ typical if it is surjective and has an infinite number critical points such that each of them is non-degenerate and all the values of $f$ at these points are pairwise different.
...
Furthermore, we show that inverse to any typical function has Riemann surface whose monodromy group coincides with finitary symmetric group $FSym(\mathbb{N})$ of permutations of naturals, which is unsolvable in the following strong sense: it does not have a normal tower of subgroups, whose factor groups are or abelian or finite. As a consequence from these facts and Topological Galois Theory, we obtain that generically (in the above sense) for $f\in E_{\rho,p}$ the solution of equation $f(w) = z$ cannot be represented via $z$ and complex constants by a finite number of the following actions: algebraic operations (i.e., rational ones and solutions of polynomial equations) and quadratures (in particular, superpositions with elementary functions)."