Closed-form expression

Summary

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.[1] 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.

Example: roots of polynomials edit

The quadratic formula

 

is a closed form of the solutions to the general quadratic equation  

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.[citation needed]

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

In higher degrees, Abel–Ruffini theorem states that there are equations whose solutions cannot be expressed in radicals, and, thus, have no closed forms. The simplest example is the equation   Galois theory provides an algorithmic method for deciding whether a particular polynomial equation can be solved in radicals.

Symbolic integration edit

Symbolic integration consists essentially of the search of closed forms for antiderivatives of functions that are specified by closed-form expressions. In this context, the basic functions used for defining closed forms are commonly logarithms, exponential function and polynomial roots. Functions that have a closed form for these basic functions are called elementary functions and include trigonometric functions, inverse trigonometric functions, hyperbolic functions, and inverse 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.

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

 

which is valid if   and   are coprime polynomials such that   is square free and  

Alternative definitions edit

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.

Analytic expression edit

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.[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.

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.[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.

Comparison of different classes of expressions edit

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 (Chow 1999) and below. A closed-form or analytic solution is sometimes referred to as an explicit solution.

Arithmetic expressions Polynomial expressions Algebraic expressions Closed-form expressions Analytic expressions Mathematical expressions
Constant Yes Yes Yes Yes Yes Yes
Elementary arithmetic operation Yes Addition, subtraction, and multiplication only Yes Yes Yes Yes
Finite sum Yes Yes Yes Yes Yes Yes
Finite product Yes Yes Yes Yes Yes Yes
Finite continued fraction Yes No Yes Yes Yes Yes
Variable No Yes Yes Yes Yes Yes
Integer exponent No Yes Yes Yes Yes Yes
Integer nth root No No Yes Yes Yes Yes
Rational exponent No No Yes Yes Yes Yes
Integer factorial No No Yes Yes Yes Yes
Irrational exponent No No No Yes Yes Yes
Exponential function No No No Yes Yes Yes
Logarithm No No No Yes Yes Yes
Trigonometric function No No No Yes Yes Yes
Inverse trigonometric function No No No Yes Yes Yes
Hyperbolic function No No No Yes Yes Yes
Inverse hyperbolic function No No No Yes Yes Yes
Root of a polynomial that is not an algebraic solution No No No No Yes Yes
Gamma function and factorial of a non-integer No No No No Yes Yes
Bessel function No No No No Yes Yes
Special function No No No No Yes Yes
Infinite sum (series) (including power series) No No No No Convergent only Yes
Infinite product No No No No Convergent only Yes
Infinite continued fraction No No No No Convergent only Yes
Limit No No No No No Yes
Derivative No No No No No Yes
Integral No No No No No Yes

Dealing with non-closed-form expressions edit

Transformation into closed-form expressions edit

The expression:

 
is not in closed form because the summation entails an infinite number of elementary operations. However, by summing a geometric series this expression can be expressed in the closed form:[2]
 

Differential Galois theory edit

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:

 
whose one antiderivative is (up to a multiplicative constant) the error function:
 

Mathematical modelling and computer simulation edit

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[3]).

Closed-form number edit

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 (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. 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 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.

Numerical computations edit

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.

Conversion from numerical forms edit

There is software that attempts to find closed-form expressions for numerical values, including RIES,[4] identify in Maple[5] and SymPy,[6] Plouffe's Inverter,[7] and the Inverse Symbolic Calculator.[8]

See also edit

References edit

  1. ^ Hyperbolic functions, inverse trigonometric functions and inverse hyperbolic functions are also allowed, since they can be expressed in terms of the preceding ones.
  2. ^ Holton, Glyn. "Numerical Solution, Closed-Form Solution". riskglossary.com. Archived from the original on 4 February 2012. Retrieved 31 December 2012.
  3. ^ Barsan, Victor (2018). "Siewert solutions of transcendental equations, generalized Lambert functions and physical applications". Open Physics. 16. De Gruyter: 232–242. doi:10.1515/phys-2018-0034. Archived from the original on Nov 3, 2023.
  4. ^ Munafo, Robert. "RIES - Find Algebraic Equations, Given Their Solution". MROB. Retrieved 30 April 2012.
  5. ^ "identify". Maple Online Help. Maplesoft. Retrieved 30 April 2012.
  6. ^ "Number identification". SymPy documentation. Archived from the original on 2018-07-06. Retrieved 2016-12-01.
  7. ^ "Plouffe's Inverter". Archived from the original on 19 April 2012. Retrieved 30 April 2012.
  8. ^ "Inverse Symbolic Calculator". Archived from the original on 29 March 2012. Retrieved 30 April 2012.

Further reading edit

External links edit