What does closed form solution usually mean?

Question

This is motivated by this question and the fact that I have no access to Timothy Chow's paper What Is a Closed-Form Number? indicated there by Qiaochu Yuan.

If an equation $f(x)=0$ has no closed form solution, what does it normally mean? Added: $f$ may depend (and normally does) on parameters.

To me this is equivalent to say that one cannot solve it for $x$ in the sense that there is no elementary expression $g(c_{1},c_{2},\ldots ,c_{p})$ consisting only of a finite number of polynomials, rational functions, roots, exponentials, logarithmic and trigonometric functions, absolute values, integer and fractional parts, such that

$f(g(c_{1},c_{2},\ldots ,c_{p}))=0$.

Answer 1

I would say it very much depends on the context, and what tools are at your disposal. For instance, telling a student who's just mastered the usual tricks of integrating elementary functions that

$$\int\frac{\exp{u}-1}{u}\mathrm{d}u$$

and

$$\int\sqrt{(u+1)(u^2+1)}\mathrm{d}u$$

have no closed form solutions is just the fancy way of saying "no, you can't do these integrals yet; you don't have the tools". To a working scientist who uses exponential and elliptic integrals, however, they do have closed forms.

In a similar vein, when we say that nonlinear equations, whether algebraic ones like $x^5-x+1=0$ or transcendental ones like $\frac{\pi}{4}=v-\frac{\sin\;v}{2}$ have no closed form solutions, what we're really saying is that we can't represent solutions to these in terms of functions that we know (and love?). (For the first one, though, if you know hypergeometric or theta functions, then yes, it has a closed form.)

I believe it is fair to say that for as long as we haven't seen the solution to an integral, sum, product, continued fraction, differential equation, or nonlinear equation frequently enough in applications to give it a standard name and notation, we just cop out and say "nope, it doesn't have a closed form".

Answer 2

To better understand closed forms, you may want to familiarize yourself with what's called Differential Algebra. Just as number theory relies on abstract structures such as rings, fields, ideals, etc. to express roots of algebraic equations using elementary numbers, similarly there is a parallel apparatus for expressing functions (i.e. solutions of differential equations) using differential rings, fields, ideals called Differential Algebra. It is this underlying mechanism that defines which functions can be expressed as "closed forms".

Parallels:

1. Similar to splitting fields for algebraic equations, there is a parallel Galois theory with Picard-Vessiot extensions and what not.
2. Similar to correspondence between subfields of number fields and Galois subgroups, on the differential side, there is a correspondence between differential subfields and subgroups of algebraic groups.
3. Just as algebraic equations can be determined to be solvable by radicals, similarly linear differential equations can be determined to be solvable by exponentials, Liouvillian functions, etc. There is an ascending tower of differential fields which can be built.

There is more... I am no expert in this differential algebra field but if you want some freely available references, see

1. Seiler Computer Algebra and differential equations
2. Van der Put Galois theory of differential equations, algebraic groups and Lie algebras
3. Papers by Michael F. Singer are good. See for example "Galois theory of linear differential equations".
4. Check the Kolchin seminar in Differential Algebra