Жабық түрдегі өрнектер және аналитикалық формулалар

Mathematical formula involving a given set of operations

In mathematics, an expression 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. Commonly, the allowed functions are nth root, exponential function, logarithm, and trigonometric functions. However, the set of basic functions depends on the context. The closed form problem arises when new ways are introduced for specifying mathematical objects, such as limits, series and integrals: given an object specified with such tools, a natural problem is to find, if possible, a closed form expression of this object, that is, an expression of this object in terms of previous ways of specifying it.

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.

An analytic expression (also known as expression in analytic form or analytic formula) is a mathematical expression constructed using well known operations that lend themselves readily to calculation. 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. 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 the Bessel functions and the gamma function are usually allowed, and often so are infinite series and continued fractions. On the other hand, limits in general, and integrals in particular, are typically excluded. 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.

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 include infinite series or continued fractions; neither includes integrals or 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 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" in the discussion of a "closed form solution", discussed in and below. A closed form or analytic solution is sometimes referred to as an explicit solution.

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).

Three subfields of the complex numbers 'C' 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 with Liouville numbers in the sense of rational approximation), EL numbers and elementary numbers. 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 '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. A narrower definition proposed in , 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. "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, Liouvillian numbers and elementary numbers contain the algebraic 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 via transcendental number 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. Some equations have no closed form solution, such as those that represent the Three body problem or the Hodgkin–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, in Maple and SymPy, Plouffe's Inverter, and the Inverse Symbolic Calculator.