What is the rigorous definition of a closed-form function? I have been wanting to ask this question for a long time. In many fields of mathematics, mathematicians are interested in whether some function $f$ is a closed-form function. However, I have never seen a rigorous definition of a closed-form function. So, has anyone in some paper or book defined rigorously what a closed-form function is? Perhaps they use an inductive definition, saying it is the least class of functions closed under certain operations. I am interested in such a definition.
 A: "Closed-form function" is not really a well-defined term. Essentially, it refers to anything that can be expressed in terms of "well-known" functions, without resorting to infinite series, integration, limits etc.  But in different contexts, various classes of functions may be considered as "well-known".
A: Dealing with the topic, I got the following impression: Because the terms "closed form" and "closed-form function" themself haven't yet been investigated as mathematical objects, there is no mathematical-scientific publication about it and therefore no standard definition exists.
Therefore their usage depends on the context and the audience. The term "closed form" means, in its non-mathematical meaning, a nice formula, appropriate for the respective audience. [Borwein/Crandell 2013] states: "Mathematics abounds in terms that are in frequent use yet are rarely made precise."
I think we shouldn't say "closed form" when we mean "elementary", because the term "elementary" is much more understandable and we should save the term "closed form" for other things. But "elementary" is, in its non-mathematical meaning, an equally unsharp term.
The term "closed form" itself concerns mathematical objects. The term "closed-form function" means functions that or whose function terms come from a set of allowed functions or function terms respectively.
[Borwein/Crandell 2013] give some answers.
If we ask for solutions in closed form, we ask for solutions that can be represented by functions from a given function class. Closed-form functions and closed-form numbers are examples of closed-form solutions. (A closed-form number is the value of a closed-form function at one argument of the function.)
Liouville defined the Elementary functions and the Liouvillian functions mathematically. So there are some decision theorems for the existence of elementary solutions (elementary functions or elementary numbers).
We have to distinguish the explicit elementary functions and the implicit elementary functions, as Chow and Khovanskii note.
With Liouville, the terms "closed form" and "closed-form function" got a second, a mathematical meaning.
We could generalize the definition and methods of the elementary functions/numbers to other kinds of closed-form functions/numbers by allowing also other functions than $\exp$ and $\ln$. (I'm working on it, and I'm still looking for collaborators.)
There are several concepts in the literature to grasp the terms "closed form" and "closed-form function". The time is ripe to collect them to a rigorous definition of their mathematical meaning and their non-mathematical meaning.
If the terms "closed form" and "closed-form function", in its mathematical meaning, are defined, they will be mathematical objects, and some general mathematical theorems, e.g. decision theorems, structure theorems and existence theorems, can be derived for them.
[Borwein/Crandell 2013] Borwein, J. M.; Crandall, R. E.: Closed Forms: What They Are and Why We Care". Notices Amer. Math. Soc. 60 (2013) (1) 50-65
