In mathematics, an expression is said to be a closed-form expression if it can be expressed analytically in terms of a bounded number of certain "well-known" functions. Typically, these well-known functions are defined to be 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).
By contrast, infinite series, integrals, limits, and infinite continued fractions are not permitted. 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 solution can be expressed as a closed-form expression. There is a subtle distinction between a "closed-form function" and a "closed-form number" in the discussion of a "closed-form solution", discussed in (Chow 1999) and below.
An area of study in mathematics is proving that no closed-form expression exists, which is referred to broadly as "Galois theory", based on the central example of closed-form solutions to polynomials.
In physics and engineering the usual terminology is "analytical solution", a solution found by evaluating functions and solving equations; systems too complex for analytical solutions can often be analysed by mathematical modelling and computer simulation.
Contents |
For example, the roots of any quadratic equation with complex coefficients can be expressed in closed form in terms of addition, subtraction, multiplication, division, and square root extraction, all elementary functions. Similarly roots of cubic and quartic (degrees three and four) can be expressed using arithmetic and roots. However, there are quintic equations without closed-form solutions using elementary functions, such as x5 − x + 1 = 0; see Galois theory.
The integral of a closed-form expression may or may not itself be expressible as a closed-form expression. This study is referred to as differential Galois theory, by analogy with algebraic Galois theory.
The basic theorem of differential Galois theory is due to Joseph Liouville in the 1830s and 1840s and hence referred to as Liouville's theorem.
A standard example of an elementary function whose antiderivative does not have a closed-form expression is e−x2, whose antiderivative is (up to constants) the error function.
Changing the definition of "well-known" to include additional functions can change the set of equations with closed-form solutions. 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. 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. 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.
Three subfields of the complex numbers C have been suggested as encoding the notion of a "closed-form number"; in increasing order of size, these are the EL numbers, Liouville numbers, and elementary numbers. The first, denoted L for Liouville numbers (not to be confused with Liouville numbers in the sense of rational approximation), is 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 in explicitly or implicitly in terms of algebraic operations, exponentials, and logarithms. A narrower definition, proposed in (Chow 1999, pp. 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 correspond to explicit 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 is transcendental. Formally, Liouville numbers and elementary numbers contain the algebraic numbers, and hence include some but not all transcendental numbers. Conversely, EL numbers do not contain all algebraic numbers, but do include some transcendental numbers. Closed-form numbers can be studied via transcendence theory, in which a major result is the Gelfond–Schneider theorem, and a major open question is Schanuel'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.