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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

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

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Can you access it from his webpage? math.mit.edu/~tchow/cv.html –  Douglas S. Stones Nov 6 '10 at 21:11
    
@Douglas S. Stones: Yes. Thank you very much. Now I can read the paper. –  Américo Tavares Nov 6 '10 at 21:22
    
@Américo Tavares: to prevent triviality you should probably require that the c_i are rational, which I think is what you meant but did not say. –  Qiaochu Yuan Nov 7 '10 at 1:19
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@Américo Tavares: 1) No, they are included. Since roots are included, i is included, so trigonometric and inverse trigonometric functions can be expressed in terms of complex exponentials and logarithms. 2) Then you are allowing f to depend on parameters, which you don't state in your original question. I was under the impression that f was a solitary function. The case that f depends on parameters is not the subject of Timothy Chow's paper, but if f is an integral it is the subject of differential Galois theory. –  Qiaochu Yuan Nov 7 '10 at 13:49
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"Closed form" is more of a general idea than a precise one. You fix some objects that you consider "elementary". Then you fix some operations that take as input collections of things, where "things" includes your elementary objects. And you ask what can you generate from that. The things you can generate are called closed-form. Usually people stick to finitary constructions even if they allow fairly complicated "elementary" functions. –  Ryan Budney Nov 7 '10 at 16:59

2 Answers 2

up vote 16 down vote accepted

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".

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I agree that "closed form" depends on context. However, I also think the default context for most people is the one defined in the question. –  John D. Cook Nov 6 '10 at 21:06
    
My copy of Abramowitz and Stegun is rather worn out from much use, which is why when explaining this stuff to other people, I always have to ask "what do you already know?" or something to that effect. I do know I may well have to say different things to a physicist and to a freshman calculus student who encounter the same integral! –  J. M. Nov 6 '10 at 21:12
    
Though anecdotes are not admissible as data, I have to say that for me personally, the reason for my being rather comfortable around these integrals is that I have had the pleasure(?) to be taught the natural logarithm as the integral of the reciprocal function. I had never encountered the natural previously at the time. I knew base-10 logarithms, and was familiar with the change-of-base formula, so finding out that this integral had the properties of a logarithm was quite the eye-opener. (cont'd) –  J. M. Nov 6 '10 at 21:17
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(cont'd) Much later, when I encountered an elliptic integral for the first time, I was all "eh, just like the logarithm..." and I was never afraid/surprised of encountering new functions. –  J. M. Nov 6 '10 at 21:20
    
+1 for all your explanations. –  Américo Tavares Nov 6 '10 at 22:10

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
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