closedform

Closed-form refers to a type of mathematical expression that can be evaluated in a finite number of standard operations and well-known functions. In practice, a closed-form expression uses constants, variables, and a finite combination of addition, subtraction, multiplication, division, exponentiation, roots, logarithms, and commonly used functions such as exponentials and trigonometric functions. Depending on the author, the allowed set of functions may also include inverse trigonometric functions or certain special functions considered elementary in that context.

A closed-form solution to a problem is one that can be written as such an expression, without recourse to iterative approximation or numerical methods. For example, the solutions to a quadratic equation ax^2 + bx + c = 0 are given by the quadratic formula, a closed-form expression. The cubic and quartic equations also have closed-form solutions in radicals, though these formulas can be lengthy. By contrast, many problems do not admit a closed-form solution; notably, Abel–Ruffini’s theorem implies that a general polynomial equation of degree five or higher does not have a solution expressible using only elementary functions in radicals.

The term is widely used across algebra, analysis, and computer algebra. It highlights a preference for explicit formulas over iterative methods when possible, because closed-form expressions are typically easier to analyze and evaluate exactly. However, what counts as “closed form” can vary by context, and some authors allow broader families of functions or accept expressions involving special functions as closed form.