# Closed Form Solution: Simple Definition, Examples

A closed form solution can be expressed in terms of mathematical operations—and functions—from a universally accepted set. Although the set may be defined differently depending on the context or mathematical fields, it’s generally understood that the number of operations and functions used must be finite.

There is a consensus over what a “closed form” is, and there is some (though less) general consensus over what are definitely not closed forms. However, there is also a grey middle area where the lines are blurred. Therefore, you should check with the text you are using (or with your instructor) to make sure your particular use doesn’t fall into this grey area.

## Examples of Closed Form Solutions

One simple example of a function with a closed form solution is the quadratic equation with complex coefficients, ax2 + bx +c = 0. Its solutions can be expressed by means of elementary functions—namely: addition, subtraction, division, multiplication, and square roots. These solutions can be expressed as:

Incidentally, cubic and quartic equations are also functions with closed form solutions, which can be expressed in terms of elementary functions.

There are certain polynomials to which, in certain contexts, no closed form solution exists. In general, equations with degree of more than 4 either don’t have a solution, or can’t be solved with simple operations. For example, Hermite polynomials have solutions involving modular elliptic functions (which Wikipedia calls “a scarcely recognizable form”).

## Alternative Definitions of Closed Form Solutions

Alternate definitions of closed form solutions have been proposed to deal with, once and for all, the problem of the ‘fuzzy definition’ usually given for closed form solution.

In a 1999 article in The American Mathematical Monthly, Timothy Chow suggests that the criteria of importance is whether functions are closed under exp and log; and defines the field EL as the intersection of all subfields which are closed under those two functions. This is a useful criteria, but has not been widely accepted as a definition for closed form.

## References

Borwein, Crandall. Closed Forms: What they are and why we care. Retrieved from https://www.carma.newcastle.edu.au/jon/closed-form.pdf on January 14, 2017
Chow, Timothy Y. What is a Closed-Form Number? The American Mathematical Monthly Vol. 106, No. 5 (May, 1999), pp. 440-448