Closed-form expression
This article needs additional citations for verification. (June 2014) |
In
The closed-form problem arises when new ways are introduced for specifying
: 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
is a closed form of the solutions to the general quadratic equation
More generally, in the context of
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
Symbolic integration
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
which is valid if and are
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 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
However, the class of expressions considered to be analytic expressions tends to be wider than that for closed-form expressions. In particular,
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 finite number of applications of well-known functions. Unlike the broader analytic expressions, the closed-form expressions do not include
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 |
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
Transformation into closed-form expressions
The expression:
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:
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 (for an example in physics, see[3]).
Closed-form number
This section may be confusing or unclear to readers. In particular, as the section is written, it seems that Liouvillian numbers and elementary numbers are exactly the same. (October 2020) |
Three subfields of the
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
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
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
- Algebraic solution– Solution in radicals of a polynomial equation
- Computer simulation – Process of mathematical modelling, performed on a computer
- Elementary function – Mathematical function
- Finitary operation– Addition, multiplication, division, ...
- Numerical solution– Study of algorithms using numerical approximation
- 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
References
- ^ Hyperbolic functions, inverse trigonometric functions and inverse hyperbolic functions are also allowed, since they can be expressed in terms of the preceding ones.
- ^ Holton, Glyn. "Numerical Solution, Closed-Form Solution". riskglossary.com. Archived from the original on 4 February 2012. Retrieved 31 December 2012.
- from the original on Nov 3, 2023.
- ^ Munafo, Robert. "RIES - Find Algebraic Equations, Given Their Solution". MROB. Retrieved 30 April 2012.
- ^ "identify". Maple Online Help. Maplesoft. Retrieved 30 April 2012.
- ^ "Number identification". SymPy documentation. Archived from the original on 2018-07-06. Retrieved 2016-12-01.
- ^ "Plouffe's Inverter". Archived from the original on 19 April 2012. Retrieved 30 April 2012.
- ^ "Inverse Symbolic Calculator". Archived from the original on 29 March 2012. Retrieved 30 April 2012.
Further reading
- Ritt, J. F. (1948), Integration in finite terms
- Chow, Timothy Y. (May 1999), "What is a Closed-Form Number?", JSTOR 2589148
- Jonathan M. Borwein and Richard E. Crandall (January 2013), "Closed Forms: What They Are and Why We Care", doi:10.1090/noti936