In mathematics, an expression (mathematics)|expression is said to be a closed-form expression if it can be expressed analytically in terms of a finite set|finite number of certain well-known function (mathematics)|functions. Typically, these well-known functions are defined to be Elementary function (differential algebra)|elementary functions—constants, one variable x, elementary operations of arithmetic (+ − × ÷), nth roots, exponent and logarithm (which thus also include trigonometric functions and inverse trigonometric functions).
Closed–form expressions are an important sub-class of analytic expressions, which contain a bounded or unbounded number of applications of well-known functions. Unlike the broader analytic expressions, the closed-form expressions do not include Series (mathematics)#Infinite series|infinite series or continued fractions; neither includes integrals or limit of a sequence|limits. Indeed, by the Stone–Weierstrass theorem, any continuous function on the unit interval can be expressed as a limit of polynomials, so any class of functions containing the polynomials and closed under limits will necessarily include all continuous functions.
Similarly, an equation or system of equations is said to have a closed-form solution if, and only if, at least one equation solving|solution can be expressed as a closed-form expression; and it is said to have an analytic solution if and only if at least one solution can be expressed as an analytic expression. There is a subtle distinction between a closed-form function and a #Closed-form number|closed-form number in the discussion of a closed-form solution, discussed in and #Closed-form number|below.
An area of study in mathematics referred to broadly as Galois theory involves proving that no closed-form expression exists in certain contexts, based on the central example of closed-form solutions to polynomials.
Equations or systems too complex for closed-form or analytical solutions can often be analysed by mathematical modelling and computer simulation.

In mathematics, an expression (mathematics)|expression is said to be a closed-form expression if it can be expressed analytically in terms of a finite set|finite number of certain well-known function (mathematics)|functions. Typically, these well-known functions are defined to be Elementary function (differential algebra)|elementary functions—constants, one variable x, elementary operations of arithmetic (+ − × ÷), nth roots, exponent and logarithm (which thus also include trigonometric functions and inverse trigonometric functions).
Closed–form expressions are an important sub-class of analytic expressions, which contain a bounded or unbounded number of applications of well-known functions. Unlike the broader analytic expressions, the closed-form expressions do not include Series (mathematics)#Infinite series|infinite series or continued fractions; neither includes integrals or limit of a sequence|limits. Indeed, by the Stone–Weierstrass theorem, any continuous function on the unit in...