Closed-form solution

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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 finite number of well-known operations. A well-known example are the two roots of a quadratic equation, which can be expressed ``in closed form'' in terms of additions, multiplications, and square root operations.

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 is well-known, since numerical implementations are widely available.

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

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