Closed-Form Solution

An equation is said to be a closed-form solution if it solves a given problem in terms of functions and mathematical operations from a given generally-accepted set. For example, an infinite sum would generally not be considered closed-form. However, the choice of what to call closed-form and what not is rather arbitrary since a new "closed-form" function could simply be defined in terms of the infinite sum.

Due to the lack of specificity in the above definition, different branches of mathematics often adopt more precise meanings of the term closed-form to apply to concepts therein. For example, within differential algebra, a function is said to be of closed-form if it is contained in some so-called Liouvillian extension field of a field , i.e., if they are obtained from rational functions by a finite sequence of adjunctions of exponentials, indefinite integrals, and algebraic functions (Churchill and Kovacic 2006). These functions are also referred to as Liouvillian (though not to be confused with the Liouville function), as well as by the even more unfortunate term "elementary".

It is worth noting that the adjective "closed" is used to describe a number of mathematical notions, e.g., the notion of closed form. Loosely speaking, a discrete function is of closed form if it shares certain essential properties with the hypergeometric function, a function which itself is defined to be the solution of the so-called hypergeometric differential equation. This particular notion of closedness is completely separate from the notion of closed-form expression as discussed above. In particular, the hypergeometric function (and hence, any closed-form function inheriting its properties) is considered a "special function" and is not expressible in terms of operations which are typically viewed as "elementary." What's more, certain agreed-upon truths like the insolvability of the quintic fail to be true if one extends consideration to a class of functions which includes the hypergeometric function, a result due to Klein (1877).