In mathematics, a closedform expression is a mathematical expression that can be evaluated in a finite number of operations. It may contain constants, variables, certain "wellknown" operations (e.g., + − × ÷), and functions (e.g., nth root, exponent, logarithm, trigonometric functions, and inverse hyperbolic functions), but usually no limit. The set of operations and functions admitted in a closedform expression may vary with author and context.
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✪ Solution 89: Closed Form of a Generating Function
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Contents
 1 Example: roots of polynomials
 2 Alternative definitions
 3 Analytic expression
 4 Comparison of different classes of expressions
 5 Dealing with nonclosedform expressions
 6 Closedform number
 7 Numerical computations
 8 Conversion from numerical forms
 9 See also
 10 References
 11 Further reading
 12 External links
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 extraction, each of which is an elementary function. For example, the quadratic equation
is tractable since its solutions can be expressed as a closedform 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 cube roots, or alternatively using arithmetic and trigonometric functions. However, there are quintic equations without closedform solutions using elementary functions, such as x^{5} − x + 1 = 0.
An area of study in mathematics referred to broadly as Galois theory involves proving that no closedform expression exists in certain contexts, based on the central example of closedform solutions to polynomials.
Alternative definitions
Changing the definition of "wellknown" to include additional functions can change the set of equations with closedform 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 wellknown since numerical implementations are widely available.
Analytic expression
An analytic expression (or expression in analytic form) is a mathematical expression constructed using wellknown operations that lend themselves readily to calculation. Similar to closedform expressions, the set of wellknown 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 closedform 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
Closedform expressions are an important subclass of analytic expressions, which contain a bounded^{[citation needed]} or an unbounded number of applications of wellknown functions. Unlike the broader analytic expressions, the closedform 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 closedform solution if, and only if, at least one solution can be expressed as a closedform 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 "closedform function" and a "closedform number" in the discussion of a "closedform solution", discussed in (Chow 1999) and below. A closedform or analytic solution is sometimes referred to as an explicit solution.
Arithmetic expressions  Polynomial expressions  Algebraic expressions  Closedform 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 
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 
Nonalgebraic root of a polynomial  No  No  No  No  Yes  Yes 
Gamma function and factorial of a noninteger  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 nonclosedform expressions
Transformation into closedform 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:^{[1]}
Differential Galois theory
The integral of a closedform expression may or may not itself be expressible as a closedform 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 closedform expression is:
whose antiderivative is (up to constants) the error function:
Mathematical modelling and computer simulation
Equations or systems too complex for closedform or analytic solutions can often be analysed by mathematical modelling and computer simulation.
Closedform number
Three subfields of the complex numbers C have been suggested as encoding the notion of a "closedform number"; in increasing order of generality, these are the Liouville numbers, EL numbers and elementary numbers. The Liouville numbers, denoted L (not to be confused with Liouville numbers in the sense of rational approximation), 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 correspond to explicit algebraic, exponential, and logarithmic operations. "EL" stands both for "ExponentialLogarithmic" and as an abbreviation for "elementary".
Whether a number is a closedform number is related to whether a number is transcendental. Formally, Liouville 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. Closedform 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.
Conversion from numerical forms
There is software that attempts to find closedform expressions for numerical values, including RIES,^{[2]} identify in Maple^{[3]} and SymPy,^{[4]} Plouffe's Inverter,^{[5]} and the Inverse Symbolic Calculator.^{[6]}
See also
 Direct method
 Algebraic solution
 Finitary operation
 Numerical solution
 Computer simulation
 Symbolic regression
 Term (logic)
 Liouvillian function
 Elementary function
References
 ^ Holton, Glyn. "Numerical Solution, ClosedForm Solution". Retrieved 31 December 2012.
 ^ Munafo, Robert. "RIES  Find Algebraic Equations, Given Their Solution". Retrieved 30 April 2012.
 ^ "identify". Maple Online Help. Maplesoft. Retrieved 30 April 2012.
 ^ "Number identification". SymPy documentation.
 ^ "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 ClosedForm Number?", American Mathematical Monthly, 106 (5): 440–448, arXiv:math/9805045, doi:10.2307/2589148, JSTOR 2589148
 Jonathan M. Borwein and Richard E. Crandall (January 2013), "Closed Forms: What They Are and Why We Care", Notices of the American Mathematical Society, 60 (1): 50–65, doi:10.1090/noti936