In mathematics, an expression or equation is in closed form if it is formed with constants, variables and a finite set of basic functions connected by arithmetic operations (+, −, ×, /, and integer powers) and function composition.
More generally, in the context of polynomial equations, a closed form of a solution is a solution in radicals; that is, a closed-form expression for which the allowed functions are only nth-roots and field operations
In fact, field theory allows showing that if a solution of a polynomial equation has a closed form involving exponentials, logarithms or trigonometric functions, then it has also a closed form that does not involve these functions.
The size of these expressions increases significantly with the degree, limiting their usefulness.
In higher degrees, the Abel–Ruffini theorem states that there are equations whose solutions cannot be expressed in radicals, and, thus, have no closed forms.
Galois theory provides an algorithmic method for deciding whether a particular polynomial equation can be solved in radicals.
Symbolic integration consists essentially of the search of closed forms for antiderivatives of functions that are specified by closed-form expressions.
The fundamental problem of symbolic integration is thus, given an elementary function specified by a closed-form expression, to decide whether its antiderivative is an elementary function, and, if it is, to find a closed-form expression for this antiderivative.
The need for logarithms and polynomial roots is illustrated by the formula which is valid if
It is possible to solve the quintic equation if general hypergeometric functions are included, although the solution is far too complicated algebraically to be useful.
[vague][citation needed] Similar to closed-form expressions, the set of well-known functions allowed can vary according to context but always includes the basic arithmetic operations (addition, subtraction, multiplication, and division), exponentiation to a real exponent (which includes extraction of the nth root), logarithms, and trigonometric functions.
On the other hand, limits in general, and integrals in particular, are typically excluded.
[citation needed] If an analytic expression involves only the algebraic operations (addition, subtraction, multiplication, division, and exponentiation to a rational exponent) and rational constants then it is more specifically referred to as an algebraic expression.
Unlike the broader analytic expressions, the closed-form expressions do not include infinite series or continued fractions; neither includes integrals or limits.
is not in closed form because the summation entails an infinite number of elementary operations.
A standard example of an elementary function whose antiderivative does not have a closed-form expression is:
Equations or systems too complex for closed-form or analytic solutions can often be analysed by mathematical modelling and computer simulation (for an example in physics, see[2]).
The Liouvillian numbers, denoted L, form the smallest algebraically closed subfield of C closed under exponentiation and logarithm (formally, intersection of all such subfields)—that is, numbers which involve explicit exponentiation and logarithms, but allow explicit and implicit polynomials (roots of polynomials); this is defined in (Ritt 1948, p. 60).
L was originally referred to as elementary numbers, but this term is now used more broadly to refer to numbers defined explicitly or implicitly in terms of algebraic operations, exponentials, and logarithms.
441–442), denoted E, and referred to as EL numbers, is the smallest subfield of C closed under exponentiation and logarithm—this need not be algebraically closed, and corresponds to explicit algebraic, exponential, and logarithmic operations.
Some equations have no closed form solution, such as those that represent the Three-body problem or the Hodgkin–Huxley model.
There is software that attempts to find closed-form expressions for numerical values, including RIES,[3] identify in Maple[4] and SymPy,[5] Plouffe's Inverter,[6] and the Inverse Symbolic Calculator.