In mathematics, an equation or system of equations is said to have a closed-form solution just in case a solution can be expressed analytically in terms of a bounded number of well-known operations. The classic example involves the two roots of a quadratic equation, which can be expressed in closed form in terms of addition and subtraction, multiplication and division, and square root extraction.

When no closed-form solutions exist – as is the case for fifth-order or higher polynomial equations, for example – such equations have to be solved numerically, typically by using some root-finding algorithm.

The precise meaning of closed-form solution depends on what operations are considered to be well-known. For example, many cumulative distribution functions cannot be expressed in closed form, unless one considers special functions such as the error function or gamma function to be well-known. For many practical computer applications, it is entirely reasonable to assume that the gamma function and other special functions are well-known, since numerical implementations are widely available.

Traditionally, the well-known functions were limited to the elementary functions. Also excluded were infinite series, limits, continued fractions, etc.

Related topics

• finitary operation
• numerical solution