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.

  • $\begingroup$ Thank you. This is exactly what I needed. I didn't know Ritt also worked on this topic. $\endgroup$ – P. Carr Aug 23 '17 at 16:57
  • $\begingroup$ Actually I could not find the short list in any of these papers. Might you be thinking of another paper? $\endgroup$ – P. Carr Aug 23 '17 at 19:11
  • $\begingroup$ Any idea for broader tags for this question?? $\endgroup$ – YCor Dec 9 '18 at 22:00
  • $\begingroup$ @P. Carr There is no short list of explicitly invertible functions. Theses functions are described by Ritt's theorem and by the proposals in my answer. A function is invertible in closed form if (iff?) it is a bijective composition of functions $f_i$ and the inverses of the $f_i$ have a closed form. That means, a function $f$ is invertible in closed form if (iff?) there exists a representation of $f$ wherein $f$ depends only on one monomial. You could honor below my answer by upvoting it. $\endgroup$ – IV_ Mar 17 '19 at 13:54

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.


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

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  • $\begingroup$ I had to smile at the last paragraph. $\endgroup$ – J. M. isn't a mathematician Aug 23 '17 at 8:18
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    $\begingroup$ The expression "closed form" (some people also call it "analytic form") is very much abused on this site. Unlike for "elementary function" there is no accepted definition. $\endgroup$ – Alexandre Eremenko Aug 23 '17 at 17:57

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 which 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.

Ritt's work Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90 seems to be the only publication which 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:


$K_0$ a field,
$X,Y_1,...,Y_n$ 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.


One can easily prove the

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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