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In mathematics, a **closed-form expression** is a mathematical expression expressed using a finite number of standard operations. It may contain constants, variables, certain "well-known" operations (e.g., + − × ÷), and functions (e.g., *n*th root, exponent, logarithm, trigonometric functions, and inverse hyperbolic functions), but usually no limit, differentiation, or integration. The set of operations and functions admitted in a closed-form expression may vary with author and context.

- Example: roots of polynomials
- Alternative definitions
- Analytic expression
- Comparison of different classes of expressions
- Dealing with non-closed-form expressions
- Transformation into closed-form expressions
- Differential Galois theory
- Mathematical modelling and computer simulation
- Closed-form number
- Numerical computations
- Conversion from numerical forms
- See also
- References
- Further reading
- External links

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 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 cube roots, or alternatively using arithmetic and trigonometric functions. However, there are quintic equations without closed-form 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 closed-form expression exists in certain contexts, based on the central example of closed-form solutions to polynomials.

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** (or **expression in analytic form**) 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 *n*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 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.

Closed-form expressions are an important sub-class of analytic expressions, which contain a bounded^{[ citation needed ]} 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 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**.

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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] }

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 antiderivative is (up to constants) the error function:

Equations or systems too complex for closed-form or analytic solutions can often be analysed by mathematical modelling and computer simulation.

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 Liouville numbers (not to be confused with Liouville numbers in the sense of rational approximation), EL numbers and elementary numbers. The **Liouville 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 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, 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. 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.

There is software that attempts to find closed-form expressions for numerical values, including RIES,^{ [2] }`identify` in Maple ^{ [3] } and SymPy,^{ [4] } Plouffe's Inverter,^{ [5] } and the Inverse Symbolic Calculator.^{ [6] }

An **algebraic number** is any complex number that is a root of a non-zero polynomial in one variable with rational coefficients. All integers and rational numbers are algebraic, as are all roots of integers. There are real and complex numbers that are not algebraic, such as π and e. These numbers are called transcendental numbers. While the set of complex numbers is uncountable, the set of algebraic numbers is countable and has measure zero in the Lebesgue measure as a subset of the complex numbers, and in this sense almost all complex numbers are transcendental.

A **complex number** is a number that can be expressed in the form *a* + *bi*, where *a* and *b* are real numbers, and *i* is a solution of the equation *x*^{2} = −1. Because no real number satisfies this equation, *i* is called an imaginary number. For the complex number *a* + *bi*, *a* is called the **real part**, and *b* is called the **imaginary part**. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world.

In mathematics, an **equation** is a statement that asserts the equality of two expressions. The word *equation* and its cognates in other languages may have subtly different meanings; for example, in French an *équation* is defined as containing one or more variables, while in English any equality is an equation.

In mathematics, an **elementary function** is a function of a single variable composed of particular simple functions.

In mathematics, a **field** is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics.

In mathematics, a **finite field** or **Galois field** is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod *p* when *p* is a prime number.

In mathematics, particularly in algebra, a **field extension** is a pair of fields such that the operations of *E* are those of *F* restricted to *E*. In this case, *F* is an **extension field** of *E* and *E* is a **subfield** of *F*. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers.

In mathematics, a **transcendental number** is a complex number that is not an algebraic number—that is, not a root of a nonzero polynomial equation with integer or equivalently rational coefficients. The best-known transcendental numbers are π and *e*.

In mathematics, **Galois theory** provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, which is in some sense simpler and better understood. It has been used to solve classic problems including showing that two problems of antiquity cannot be solved as they were stated ; showing that there is no quintic formula; and showing which polygons are constructible.

In symbolic computation, at the intersection of mathematics and computer science, the **Risch algorithm** is an algorithm for indefinite integration. It is used in some computer algebra systems to find antiderivatives. It is named after the American mathematician Robert Henry Risch, a specialist in computer algebra who developed it in 1968.

Field theory is the branch of mathematics in which fields are studied. This is a glossary of some terms of the subject.

A **transcendental function** is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function. In other words, a **transcendental function** "transcends" algebra in that it cannot be expressed in terms of a finite sequence of the algebraic operations of addition, multiplication, and root extraction.

In mathematics, an **algebraic equation** or **polynomial equation** is an equation of the form

In mathematics, an **algebraic function** is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power. Examples of such functions are:

**Transcendental number theory** is a branch of number theory that investigates transcendental numbers, in both qualitative and quantitative ways.

An **algebraic solution** or **solution in radicals** is a closed-form expression, and more specifically a closed-form algebraic expression, that is the solution of an algebraic equation in terms of the coefficients, relying only on addition, subtraction, multiplication, division, raising to integer powers, and the extraction of nth roots.

In mathematics, an **algebraic expression** is an expression built up from integer constants, variables, and the algebraic operations. For example, 3*x*^{2} − 2*xy* + *c* is an algebraic expression. Since taking the square root is the same as raising to the power 1/2,

In mathematics, **Liouville's theorem**, originally formulated by Joseph Liouville in 1833 to 1841, places an important restriction on antiderivatives that can be expressed as elementary functions.

**Algebra** (from Arabic: الجبر is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. It includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra; the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians.

In mathematics, the **Liouvillian functions** comprise a set of functions including the elementary functions and their repeated integrals. Liouvillian functions can be recursively defined as integrals of other Liouvillian functions.

- ↑ Holton, Glyn. "Numerical Solution, Closed-Form Solution". Archived from the original on 4 February 2012. 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.

- Ritt, J. F. (1948),
*Integration in finite terms* - Chow, Timothy Y. (May 1999), "What is a Closed-Form 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

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