Inmathematics, anexpression orequation is inclosed form if it is formed withconstants,variables and afinite set of basicfunctions connected by arithmetic operations (+, −, ×, /, andinteger powers) andfunction composition. Commonly, the allowed functions arenth root,exponential function,logarithm, andtrigonometric functions.^[a] However, the set of basic functions depends on the context.

Theclosed-form problem arises when new ways are introduced for specifyingmathematical objects, such aslimits,series andintegrals: given an object specified with such tools, a natural problem is to find, if possible, aclosed-form expression of this object, that is, an expression of this object in terms of previous ways of specifying it.

Thequadratic formula

${\displaystyle x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}.}$ 

is aclosed form of the solutions to the generalquadratic equation${\displaystyle a{x}^2+bx+c=0.}$ 

More generally, in the context ofpolynomial equations, a closed form of a solution is asolution in radicals; that is, a closed-form expression for which the allowed functions are onlynth-roots and field operations${\displaystyle (+,-,\times ,/).}$  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.^{[citation needed]}

There are expressions in radicals for all solutions ofcubic equations (degree 3) andquartic equations (degree 4). The size of these expressions increases significantly with the degree, limiting their usefulness.

In higher degrees, theAbel–Ruffini theorem states that there are equations whose solutions cannot be expressed in radicals, and, thus, have no closed forms. A simple example is the equation${\displaystyle x^5-x-1=0.}$ Galois theory provides analgorithmic method for deciding whether a particular polynomial equation can be solved in radicals.

Symbolic integration consists essentially of the search of closed forms forantiderivatives of functions that are specified by closed-form expressions. In this context, the basic functions used for defining closed forms are commonlylogarithms,exponential function andpolynomial roots. Functions that have a closed form for these basic functions are calledelementary functions and includetrigonometric functions,inverse trigonometric functions,hyperbolic functions, andinverse hyperbolic functions.

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.

Forrational functions; that is, for fractions of twopolynomial functions; antiderivatives are not always rational fractions, but are always elementary functions that may involve logarithms and polynomial roots. This is usually proved withpartial fraction decomposition. The need for logarithms and polynomial roots is illustrated by the formula

${\displaystyle \int \frac{f(x)}{g(x)}dx=\sum _{\alpha \in \mathrm{Roots}(g(x))}\frac{f(\alpha )}{g^{\prime }(\alpha )}\ln (x-\alpha ),}$ 

which is valid if${\displaystyle f}$  and${\displaystyle g}$  arecoprime polynomials such that${\displaystyle g}$  issquare free and 

Changing the definition of "well known" to include additional functions can change the set of equations with closed-form solutions. Manycumulative distribution functions cannot be expressed in closed form, unless one considersspecial functions such as theerror function orgamma function to be well known. It is possible to solve the quintic equation if generalhypergeometric functions are included, although the solution is far too complicated algebraically to be useful. 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.

Ananalytic expression (also known asexpression in analytic form oranalytic formula) is amathematical expression constructed using well-known operations that lend themselves readily to calculation.^{[vague][citation needed]} Similar to closed-form expressions, the set of well-known functions allowed can vary according to context but always includes thebasic arithmetic operations (addition,subtraction, multiplication, and division), exponentiation to a real exponent (which includes extraction of thenth root), logarithms, and trigonometric functions.

However, the class of expressions considered to be analytic expressions tends to be wider than that for closed-form expressions. In particular,special functions such as theBessel functions and thegamma function are usually allowed, and often so areinfinite series andcontinued fractions. On the other hand,limits in general, andintegrals 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 analgebraic expression.

Closed-form expressions are an important sub-class of analytic expressions, which contain a finite number of applications of well-known functions. Unlike the broader analytic expressions, the closed-form expressions do not includeinfinite series orcontinued fractions; neither includesintegrals orlimits. Indeed, by theStone–Weierstrass theorem, anycontinuous function on theunit 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, anequation orsystem of equations is said to have aclosed-form solutionif and only if at least onesolution can be expressed as a closed-form expression; and it is said to have ananalytic solution if and only if at least one solution can be expressed as an analytic expression. There is a subtle distinction between a "closed-formfunction" and a "closed-formnumber" in the discussion of a "closed-form solution", discussed in (Chow 1999) andbelow. A closed-form or analytic solution is sometimes referred to as anexplicit solution.

The expression:$${\displaystyle f(x)=\sum _{n=0}^{\mathrm{\infty }}\frac{x}{2^n}}$$ is not in closed form because the summation entails an infinite number of elementary operations. However, by summing ageometric series this expression can be expressed in the closed form:^[1]$${\displaystyle f(x)=2x.}$$ 

The integral of a closed-form expression may or may not itself be expressible as a closed-form expression. This study is referred to asdifferential Galois theory, by analogy with algebraic Galois theory.

The basic theorem of differential Galois theory is due toJoseph Liouville in the 1830s and 1840s and hence referred to asLiouville's theorem.

A standard example of an elementary function whose antiderivative does not have a closed-form expression is:$${\displaystyle e^{-x^2},}$$  whose one antiderivative is (up to a multiplicative constant) theerror function:$${\displaystyle \mathrm{erf}(x)=\frac{2}{\sqrt{\pi }}{\int }_0^x{e}^{-t^2}dt.}$$ 

Equations or systems too complex for closed-form or analytic solutions can often be analysed bymathematical modelling andcomputer simulation (for an example in physics, see^[2]).

Three subfields of thecomplex numbersC have been suggested as encoding the notion of a "closed-form number"; in increasing order of generality, these are the Liouvillian numbers (not to be confused withLiouville numbers in the sense of rational approximation), EL numbers andelementary numbers. TheLiouvillian numbers, denotedL, form the smallestalgebraically closed subfield ofC closed under exponentiation and logarithm (formally, intersection of all such subfields)—that is, numbers which involveexplicit exponentiation and logarithms, but allow explicit andimplicit polynomials (roots of polynomials); this is defined in (Ritt 1948, p. 60).L was originally referred to aselementary 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. A narrower definition proposed in (Chow 1999, pp. 441–442), denotedE, and referred to asEL numbers, is the smallest subfield ofC closed under exponentiation and logarithm—this need not be algebraically closed, and corresponds toexplicit algebraic, exponential, and logarithmic operations. "EL" stands both for "exponential–logarithmic" and as an abbreviation for "elementary".

Whether a number is a closed-form number is related to whether a number istranscendental. Formally, Liouvillian numbers and elementary numbers contain thealgebraic numbers, and they include some but not all transcendental numbers. In contrast, EL numbers do not contain all algebraic numbers, but do include some transcendental numbers. Closed-form numbers can be studied viatranscendental number theory, in which a major result is theGelfond–Schneider theorem, and a major open question isSchanuel's conjecture.

For purposes of numeric computations, being in closed form is not in general necessary, as many limits and integrals can be efficiently computed. Some equations have no closed form solution, such as those that represent theThree-body problem or theHodgkin–Huxley model. Therefore, the future states of these systems must be computed numerically.

There is software that attempts to find closed-form expressions for numerical values, including RIES,^[3]identify inMaple^[4] andSymPy,^[5] Plouffe's Inverter,^[6] and theInverse Symbolic Calculator.^[7]

• Algebraic solution – Solution in radicals of a polynomial equationPages displaying short descriptions of redirect targets
• Computer simulation – Process of mathematical modelling, performed on a computer
• Elementary function – A kind of mathematical function
• Finitary operation – Addition, multiplication, division, ...Pages displaying short descriptions of redirect targets
• Numerical solution – Methods for numerical approximationsPages displaying short descriptions of redirect targets
• Liouvillian function – Elementary functions and their finitely iterated integrals
• Symbolic regression – Type of regression analysis
• Tarski's high school algebra problem – Mathematical problem
• Term (logic) – Components of a mathematical or logical formula
• Tupper's self-referential formula – Formula that visually represents itself when graphed

• Ritt, J. F. (1948),Integration in finite terms
• Chow, Timothy Y. (May 1999), "What is a Closed-Form Number?",American Mathematical Monthly,106 (5):440–448,arXiv:math/9805045,doi:10.2307/2589148,JSTOR 2589148
• Jonathan M. Borwein and Richard E. Crandall (January 2013), "Closed Forms: What They Are and Why We Care",Notices of the American Mathematical Society,60 (1):50–65,doi:10.1090/noti936

• Weisstein, Eric W."Closed-Form Solution".MathWorld.
• Closed-form continuous-time neural networks