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My initial question contained about how to classify functions whose integrals and inverses do not have a closed form. But I found this question: How can you prove that a function has no closed form integral?

So this does satisfy my first part to the question but inverse still remains a question. The only thing I was able to think was that mostly(though this does not include a lot of function, for e.g.: $\sin(x)+\cos(x)$ ) the functions which have addition as the operation . E.g.: $f(x)=\sin(x)+e^x$ and many more.

So the question is: How to classify all functions whose inverse does not have a closed form, that is cannot be written in terms of standard mathematical functions. Here the standard mathematical function include all the functions that were mentioned in the question linked above.

[edit]: I was told that the idea might be that I would have to find a differential equation that $f^{-1}(x)$ and then use Differential Galois theory to prove it is not elementary. So I took the example I had given in the question. But I am stuck. If I do not know how to find the inverse how should I use it to get a differential equation. So I tried finding a differential equation for $f(x)=\sin(x)+e^x$ itself but that was also a failed attempt as I was not able to find any differential equation satisfying $f(x)$ as solution of it. So what is happening here?

IV_
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Sayan Chattopadhyay
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1 Answers1

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If you use the term "closed form", you have to declare which kinds of functions you will allow.

Let us answer your question for the Elementary functions instead. The elementary functions are a special kind of closed-form expressions. If $f$ is an elementary function, the following statements are equivalent:

  • $f$ is generated from its only argument variable in a finite number of steps by performing only arithmetic operations, power functions with integer exponents, root functions, exponential functions, logarithm functions, trigonometric functions, inverse trigonometric functions, hyperbolic functions and/or inverse hyperbolic functions.

  • $f$ is generated from its only argument variable in a finite number of steps by performing only arithmetic operations, exponentials and/or logarithms.

  • $f$ is generated from its only argument variable in a finite number of steps by performing only explicit algebraic functions, exponentials and/or logarithms

  • $f$ is a composition of a finite number of only $\exp$, $\ln$ and/or unary or multiary univalued algebraic functions.

The answer to your question is given by the incomprehensibly unfortunately hardly noticed theorem of Joseph Fels Ritt in Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90: An elementary function which has an elementary inverse is a composition of a finite number of only $\exp$, $\ln$ and/or unary algebraic functions.

This theorem of Ritt shows that no antiderivatives, no differentiation and no differential fields are needed for defining the Elementary functions.

Ritt's theorem is proved also in Risch, R. H.: Algebraic Properties of the Elementary Functions of Analysis. Amer. J. Math. 101 (1979) (4) 743-759.

A better way is to ask which kinds of functions have partial closed-form inverses. It is easy to prove a theorem that is in a certain sense opposite to Ritt's theorem: If $f$ is a function with $f=f_1\circ\ldots\circ f_n$, where $n\in\mathbb{N}_{\ge 1}$ and $\forall i\in\{1,...n\}\colon f_i\colon D_i\subseteq\mathbb{C}^{k_i}\to\mathbb{C}^{k_i}$, for each partial inverse $\phi$ of $f$, $\phi=\phi_n\circ\ldots\circ \phi_1$ holds, where $\forall i\in\{1,...n\}\colon \phi_i$ is a partial inverse of $f_i$.

According to Ritt's theorem, this are the only elementary functions with elementary inverses. It is not known if my previous sentence is true also for other classes of functions except the Elementary functions.

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