Closed-Form Solution

A closed-form solution is a mathematical expression that 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 it is 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). Liouvillian functions include indefinite integrals of elementary functions and form a broader class than elementary functions.

The adjective "closed" also describes unrelated mathematical notions, including closed form. In discrete summation, a hypergeometric term has ratios that are rational functions of its integer arguments under shifts. This differs from an analytic hypergeometric function, which solves the hypergeometric differential equation. Neither class is universally non-elementary. For example, the constant discrete term is elementary, and for . Analytic hypergeometric functions are generally special functions. Allowing them permits representations of solutions of the general quintic equation, which cannot in general be expressed by radicals (Klein 1877).