In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a closed-form expression is a
mathematical expression that uses a
finite number of standard operations. It may contain
constants
Constant or The Constant may refer to:
Mathematics
* Constant (mathematics), a non-varying value
* Mathematical constant, a special number that arises naturally in mathematics, such as or
Other concepts
* Control variable or scientific const ...
,
variables, certain well-known
operations (e.g., + − × ÷), and
functions (e.g.,
''n''th root,
exponent,
logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
,
trigonometric functions, and
inverse hyperbolic functions
In mathematics, the inverse hyperbolic functions are the inverse functions of the hyperbolic functions.
For a given value of a hyperbolic function, the corresponding inverse hyperbolic function provides the corresponding hyperbolic angle. T ...
), but usually no
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
,
differentiation, or
integration. The set of operations and functions may vary with author and context.
Example: roots of polynomials
The solutions of any
quadratic equation with
complex coefficients can be expressed in closed form in terms of
addition,
subtraction,
multiplication,
division, and
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
...
extraction, each of which is an
elementary function. For example, the quadratic equation
:
is tractable since its solutions can be expressed as a closed-form expression, i.e. in terms of elementary functions:
:
Similarly, solutions of cubic and quartic (third and fourth degree) equations can be expressed using arithmetic, square roots, and
th roots. However, there are
quintic equations without such closed-form solutions, for example ; this is
Abel–Ruffini theorem.
The study of the existence of closed forms for
polynomial roots is the initial motivation and one of the main achievements of the area of mathematics named
Galois theory.
Alternative definitions
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
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers excep ...
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
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
th 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
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrar ...
and the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers excep ...
are usually allowed, and often so are
infinite series and
continued fractions. On the other hand,
limits in general, and
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
s 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.
Comparison of different classes of expressions
Closed-form expressions are an important sub-class of analytic expressions, which contain a bounded or an unbounded 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
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
s or
limits. Indeed, by the
Stone–Weierstrass theorem
In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval can be uniformly approximated as closely as desired by a polynomial function. Because polynomials are among the ...
, 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
In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought. An equation system is usually classified in the same manner as single ...
is said to have a closed-form solution if, and only if, at least one
solution
Solution may refer to:
* Solution (chemistry), a mixture where one substance is dissolved in another
* Solution (equation), in mathematics
** Numerical solution, in numerical analysis, approximate solutions within specified error bounds
* Solutio ...
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
Below may refer to:
*Earth
* Ground (disambiguation)
*Soil
*Floor
* Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Ernst von Below (1863–1955), German World War I general
*Fred Below ...
. A closed-form or analytic solution is sometimes referred to as an explicit solution.
Dealing with non-closed-form expressions
Transformation into closed-form expressions
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:
Differential 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:
whose one antiderivative is (
up to a multiplicative constant) the
error function:
Mathematical modelling and computer simulation
Equations or systems too complex for closed-form or analytic solutions can often be analysed by
mathematical modelling and
computer simulation.
Closed-form number
Three subfields of the complex numbers 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 , form the smallest ''
algebraically closed'' subfield of 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 . 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 , and referred to as EL numbers, is the smallest subfield of 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, 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
In mathematics, the Gelfond–Schneider theorem establishes the transcendence of a large class of numbers.
History
It was originally proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider.
Statement
: If ''a'' and ''b'' a ...
, and a major open question is
Schanuel's conjecture.
Numerical computations
For purposes of numeric computations, being in closed form is not in general necessary, as many limits and integrals can be efficiently computed.
Conversion from numerical forms
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.
See also
*
*
*
*
*
*
*
*
*
*
References
Further reading
*
*
*
External links
*
Closed-form continuous-time neural networks
{{DEFAULTSORT:Closed-Form Expression
Algebra
Special functions