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

To me this is equivalent to say that one cannot solve it for x x in the sense that there is no elementary expression g(c1,c2Boot Men's Caterpillar Brown Regulator Toe Work Steel WXWdHaq,,cp) g(c1,c2,,cp) 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(c1,Cork 10 Shoe Sandals Summer Patchwork Beach Flats Buckle Casual 2018 Slipper Double New Women Perfues naFOxE6w6c2Laced Boots Industrial Brown MR86304 Mens Light McRae M 8 Western ,,cp))=0 f(g(c1,c2,,cp))=0 .

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

expu1udu

expuLight 8 M McRae Western Mens Laced Industrial Brown Boots MR86304 1udu

and

(u+1)(u2+1)du

(u+1)(u2+1)du

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 x5x+1=0 x5x+1=0 or transcendental ones like π4=vsinv2 π4=vsinv2Boots Mens Light Industrial 8 McRae Brown MR86304 M Western Laced 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".

  • 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. is not a mathematician 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. is not a mathematician Nov 6 '10 at 21:17
  • 3
    (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. is not a mathematician Bows with Heels Pull WeiPoot Soft Low Toe Women's top Round Closed on High Boots Material Black wnO6tan
  • @ John D. Cook I disagree. Fractional and integer parts are not default closed forms for me. On the other hand, factorial and Gamma function are included –  Anixx Dec 15 '10 at 2:08

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

Closed form solution is a solution that can be represented without using limits, and as such, integrals, infinite sums, derivatives. Only using functions, their compositions and arithmetic operations on them. The class of functions allowed may vary. By default it usually allows elementary functions plus Gamma function, plus Polygamma function, plus Hurwitz zeta function. Sometimes, service function like "integer part", "absolute value", "argument", "real part", "imaginary part" may be allowed.

Let us assume, f(x)=0 f(x)=0 is to be solved for x x .

If an equation f(x)=0 f(x)=0M Industrial Mens Laced Light MR86304 McRae 8 Boots Brown Western has no closed-form solution, the equation has no solution which can be expressed as a closed-form expression.

A mathematical expression is a closed-form expression iff it contains only finite numbers of only constants, functions, operations and/or variables.

Sensefully, all the constants, functions and operations in a given closed-form expression should be from given sets.

Let us say, a (local) closed-form inverse (f1 f1 ) is a (local) inverse (= inverse function) which can be expressed as closed-form expression.

Because of f(x)=0 fBrown Light Mens Boots Industrial M Western MR86304 McRae 8 Laced (x)=0 and the definition of a (local) inverse f1(f(x))=x f1(f(x))=x , the following holds: f1(f(x))=f1(0) f1(f(x))=f1(0) , x=f1(0) x=f1(0) . And therefore: If an equation f(x)=0 f(x)=0 has no closed-form solution, the function f f has no local closed-form inverse, or a local closed-form inverse exists but is not defined for the argument 0 0 of the right side of the equation. This means, x x cannot be isolated on only one side of the equation

  • by applying a local closed-form inverse,

  • by only applying the local closed-form inverses and inverse operations of the closed-form functions respective operations which are contained in the expression f(x) f(x) .

The existence of a local closed-form inverse is a sufficient but not a necessary criterion for the existence of a closed-form solution.

The elementary functions are a special kind of closed-form expressions. If f f is an elementary function, the following statements are equivalent:

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

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

Whereas Joseph Fels Ritt allows explicit and implicit algebraic functions, Timothy Chow restricts the approved algebraic operations to the explicit algebraic functions, that are the arithmetic operations.

My take on this question, from a practical standpoint:

In the world of computers, there are no "closed forms."

"Closed form" is a mathematician's label for certain mathematical expressions which he deems "elementary." More specifically, it's a way of saying, "We don't care about the algorithms for evaluating this expression."

What makes a "closed form" is that the algorithmic steps involved in computing it are regarded in algebraic manipulation as one atomic step.


A classic example of this is the factorial, n! n! . Strictly speaking, the definition involves a recurrence. However, it comes up in practice so often, mathematicians label it a "closed form" itself.

Now, perhaps some clever mathematical algorithms expert might come up with a more efficient way to compute the factorial of arbitrary values of n n , which does not involve actually performing n1 n1 multiplication steps. The point I'm making is that the label "closed form" doesn't depend on that better algorithm being known, or even being possible. It just means, "We are regarding the algorithm for computing this as not a crucial question (for the current text)."


In fact, if you get right down to it, f(x)=x+1 f(x)=x+1 is the ultimate closed form. The unary "increment" operator.

Adding bigger numbers (the binary "sum" operator) can be seen as a "recurrence" or repeated application of the "increment" operator.

Multiplication itself is repeated application of the "sum" operator, and likewise exponentiation is a repeated application of multiplication.

But all of these are considered as "closed forms." The concept is not a fixed one.


To quote Concrete Mathematics:

We could give a rough definition like this: An expression for a quantity f(n) is in closed form if we can compute it using at most a fixed number of “well known” standard operations, independent of n. For example, 2n – 1 and n(n + 1)/2 are closed forms because they involve only addition, subtraction, multiplication, division, and exponentiation, in explicit ways.

The total number of simple closed forms is limited, and there are recurrences that don’t have simple closed forms. When such recurrences turn out to be important, because they arise repeatedly, we add new operations to our repertoire; this can greatly extend the range of problems solvable in “simple” closed form. For example, the product of the first n integers, n!, has proved to be so important that we now consider it a basic operation. The formula ‘n!’ is therefore in closed form, although its equivalent ‘1·2·. . .·n’ is not.

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One usually refers to "closed form" as a solution involving functions we commonly know of. This class of functions varies from problem to problem, field to field. For example, one might say x5x+1=0 x5x+1=0 has no closed form solution because x x can't be solved for in terms of radicals. However, it can be solved in terms of Bring radicals. Here, closed form might mean "a combination of rational numbers, addition, multiplication, and radicals".

One could extend "closed form" to mean a solutions involving elementary functions, which are functions made of a finite composition of arithmetic operations, exponentials, logarithms, constants, and solutions to algebraic equations. Under this definition of closed form, x5x+1=a x5x+1=a has a closed form solution since it is an algebraic equation.

However, upon entering calculus, you will find there are many integrals you cannot solve. For example,

1+x3 dx, ex2 dx

1+x3 dx, ex2 dx

These are antiderivatives of elementary functions, though they themselves are not elementary. One can make these integrals as "closed form" by having it be the class of Liouvillian functions, which are elementary functions and their antiderivatives.


So as your problems get harder and your field changes, closed form will have a different meaning to you (and not limited to the above).

For most purposes, I think closed form is implied to mean elementary function though.

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