Proving that special functions do not have closed-form expression When dealing with special functions, like Erf, one should encounter the following statement 

This function cannot be expressed in terms of classical functions

This seems pretty true, but I was wondering why. Could you prove it by hand ? I found the following definition on Wikipedia :

In mathematics, an expression is said to be a closed-form expression if it can be expressed analytically in terms of a finite number of certain "well-known" functions. Typically, these well-known functions are defined to be elementary functions—constants, one variable x, elementary operations of arithmetic (+ − × ÷), nth roots, exponent and logarithm (which thus also include trigonometric functions and inverse trigonometric functions).

I am not just interested in the error function but more generally in any special function. I was wondering if any "field" of mathematics related to this question existed. Any interesting reference of link would be appreciated !
 A: To be a bit more precise, the part of mathematics concerned with closed-form integrals and closed form solutions to ODE is called "Differential Algebra." This is not for the faint-hearted, and was begun by RITT and his student KOLCHIN at Columbia. One of the current leaders in this study is called "The Bozh" by my friend Dmitry, anyway Michael Boshernitzan  of Rice. I have a book by Kaplansky called, and I think this is clever, "Differential Algebra." 
A differential field is simply one which has a derivation, $D(uv) = u D(v) + v D(u).$ One can then decide whether a new item, an indefinite integral especially, is in the field. Thus the meaning of the phrase "closed form" is entirely up to the individual mathematician: a function is elementary if it is in the differential field being considered. Put another way, a function is elementary if you say it is. In the same way, if a child brings home a stray puppy, the parents tell the child not to give it a name as it will be gone soon. Your elementary functions are just those to which you have assigned a name.  
There is a side note here: modern computer algebra systems use Ritt's algorithm, and many types of improvements based on including Grobner bases into the mix, for deciding whether some integral has a closed form, as far as the CAS is concerned. Note that one may do this in steps, first decide whether it is in the traditional basket of elementary functions we learn in high-school, next add in a few less traditional, see if that changes things. 
There is a type of needlework called CROSS STITCH, for which a sampler is a piece small enough to be framed and hung on a wall. A book with text examples at BOOK.  We could commission such a thing and send it to J.M. It would read 
$$\begin{array}{c}\text{Elementary is in the}\cr\text{Eye of the Beholder}\end{array}$$
or maybe
$$\begin{array}{c}\text{Closed Form is in the}\cr\text{Eye of the Beholder}\end{array}$$
A: The "field" of mathematics that deals with such a question is, coincidentally, field theory. A function has a closed-form representation if and only if it belongs to a certain tower of function fields.
In practice, given a function $f: \mathbb{F}\to\mathbb{F}$, we want to determine if there exists an $n\in\mathbb{N}$ and a sequence of fields $\{K_i\}_{i=0}^{n}$ such that:


*

*$K_{i+1}$ is a simple field extension $K_i$ (described below)

*$K_0\equiv \mathbb{F[x]}$

*$f\in K_n$


The field extension must be of the form $K_{i+1}=K_i[g]$ where either:


*

*$g$ is algebraic over $K_i$

*$g=\exp(h)$ for $h\in K_i$

*$g=\ln(h)$ for $h\in K_i$


An elementary introduction can be found here
A: A closed-form function is a function from a given set of functions where the functions themselves or the generator of the set is given.
Often, a function is found as result of series or a solution of an equation, a differential equation, an integral equation, an integro-differential equation, an operator equation, a functional equation or of a system of such equations. One can ask if a given solution can be generated by functions from a given class of functions. But results are only rare. Results are available for the Elementary functions and for the Liouvillian functions.
To apply Differential Algebra, you have to choose a differential field and its differential extensions. The Elemetary functions according to Liouville and Ritt form a differential field. These elementary functions are those functions which are obtained in a finite number of steps by performing algebraic operations and taking exponentials and
logarithms. The Liouvillian functions form also a differential field. A Liouvillian function is an elementary function or (recursively) the integral of a Liouvillian function, e.g. the nonelementary integral of an elementary function. To decide if a given Special function is in your given differential field, you can investigate some differential equations of your Special function and apply Liouville's theorem in Differential Algebra. This is done in Rosenlicht, M.: On the explicit solvability of certain transcendental equations. Publications mathématiques de l'IHÉS 36 (1969) 15-22.
Another method to decide if a given Special function is an Elementary function according to Liouville and Ritt is to investigate if it is an elementary inverse of an elementary function. This can be done by the main theorem in Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90 that is proved also in Risch, R. H.: Algebraic Properties of the Elementary Functions of Analysis. Amer. J. Math. 101 (1979) (4) 743-759.
Another method is decribed in Khovanskii, A.: Topological Galois Theory. Solvability and Unsolvability of Equations in Finite Terms. Springer 2014 and articles of A. Khovanskii and Y. Burda. It is applied in the following article.
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
