Closed-form expression

In mathematics, a closed-form expression is a mathematical expression that can be evaluated in a finite number of operations. It may contain constants, variables, certain "well-known" operations (e.g., + − × ÷), and functions (e.g., nth root, exponent, logarithm, trigonometric functions, and inverse hyperbolic functions), but usually no limit. The set of operations and functions admitted in a closed-form expression may vary with author and context.

Problems are said to be tractable if they can be solved in terms of a closed-form expression.

Example: roots of polynomials

The solutions of any quadratic equation with complex coefficients can be expressed in closed form in terms of addition, subtraction, multiplication, division, and square root extraction, each of which is an elementary function. For example, the quadratic equation:

is tractable since its solutions can be expressed as closed-form expression, i.e. in terms of elementary functions:

Similarly solutions of cubic and quartic (third and fourth degree) equations can be expressed using arithmetic, square roots, and cube roots, or alternatively using arithmetic and trigonometric functions. However, there are quintic equations without closed-form solutions using elementary functions, such as x⁵ − x + 1 = 0.