Elementary function (differential algebra)

In mathematics

Mathematics

Mathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....

, an elementary function is a function

Function (mathematics)

In mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...

 built from a finite number of exponential

Exponential function

In mathematics, the exponential function is the function e^x, where e is the number such that the function e^x equals its own derivative. The exponential function is used to model phenomena when a constant change in the independent variable gives the same proportional change ...

s, logarithm

Logarithm

In mathematics, the logarithm of a number to a given base is the power or exponent to which the base must be raised in order to produce the number....

s, constant

Coefficient

In mathematics, a coefficient is a constant multiplicative factor of a specific object. For example, in the expression 9x², the coefficient of x² is 9.The object can be such things as a variable, a vector, a function, etc...

s, one variable

Variable (mathematics)

A variable is a symbol that stands for a value that may vary; the term usually occurs in opposition to constant, which is a symbol for a non-varying value, i.e. completely fixed or fixed in the context of use...

, and nth roots through composition

Function composition

In mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...

 and combinations using the four elementary operations

Arithmetic

Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations, such as addition, subtraction, multiplication and division...

 (+ – × ÷). By allowing these functions (and constants to be complex numbers, trigonometric function

Trigonometric function

In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...

s and their inverses

Inverse trigonometric function

In mathematics, the inverse trigonometric functions or cyclometric functions are the so-called inverse functions of the trigonometric functions, though they do not meet the official definition for inverse functions as their ranges are subsets of the domains of the original functions...

 are included in the elementary functions (see Trigonometric function#Relationship to exponential function and complex numbers).

The roots of equations are the functions implicitly defined as solving a polynomial equation with constant coefficients.

In mathematics

Mathematics

Mathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....

, an elementary function is a function

Function (mathematics)

In mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...

 built from a finite number of exponential

Exponential function

In mathematics, the exponential function is the function e^x, where e is the number such that the function e^x equals its own derivative. The exponential function is used to model phenomena when a constant change in the independent variable gives the same proportional change ...

s, logarithm

Logarithm

In mathematics, the logarithm of a number to a given base is the power or exponent to which the base must be raised in order to produce the number....

s, constant

Coefficient

In mathematics, a coefficient is a constant multiplicative factor of a specific object. For example, in the expression 9x², the coefficient of x² is 9.The object can be such things as a variable, a vector, a function, etc...

s, one variable

Variable (mathematics)

A variable is a symbol that stands for a value that may vary; the term usually occurs in opposition to constant, which is a symbol for a non-varying value, i.e. completely fixed or fixed in the context of use...

, and nth roots through composition

Function composition

In mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...

 and combinations using the four elementary operations

Arithmetic

Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations, such as addition, subtraction, multiplication and division...

 (+ – × ÷). By allowing these functions (and constants to be complex numbers, trigonometric function

Trigonometric function

In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...

s and their inverses

Inverse trigonometric function

In mathematics, the inverse trigonometric functions or cyclometric functions are the so-called inverse functions of the trigonometric functions, though they do not meet the official definition for inverse functions as their ranges are subsets of the domains of the original functions...

 are included in the elementary functions (see Trigonometric function#Relationship to exponential function and complex numbers).

The roots of equations are the functions implicitly defined as solving a polynomial equation with constant coefficients. For polynomials of degree four and smaller there are explicit formulas for the roots (the formulas are elementary functions), but even for higher degree polynomials the fundamental theorem of algebra

Fundamental theorem of algebra

In mathematics, the fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root...

 and the implicit function theorem

Implicit function theorem

In the branch of mathematics called multivariable calculus, the implicit function theorem is a tool which allows relations to be converted to functions. It does this by representing the relation as the graph of a function. There may not be a single function whose graph is the entire relation, but...

 assures the existence of a function that returns each one of the roots of a polynomial equation.

Elementary functions were introduced by Joseph Liouville

Joseph Liouville

Joseph Liouville was a French mathematician.- Life and work :Liouville graduated from the École Polytechnique in 1827. After some years as an assistant at various institutions including the Ecole Centrale Paris, he was appointed as professor at the École Polytechnique in 1838...

 in a series of papers from 1833 to 1841. An algebraic treatment of elementary functions was started by Joseph Fels Ritt in the 1930s.

Examples

Examples of elementary functions include:
and
The domain of this last function does not include any real number. An example of a function that is not elementary is the error function

Error function

In mathematics, the error function is a special function of sigmoid shape which occurs in probability, statistics, materials science, and partial differential equations...

a fact that cannot be seen directly from the definition of elementary function but can be proven using the Risch algorithm

Risch algorithm

The Risch algorithm, named after Robert H. Risch, is an algorithm for the calculus operation of indefinite integration . The algorithm transforms the problem of integration into a problem in algebra. It is based on the form of the function being integrated and on methods for integrating rational...

.

Differential algebra

The mathematical definition of an elementary function, or a function in elementary form, is considered in the context of differential algebra

Differential algebra

In mathematics, differential rings, differential fields and differential algebras are rings, fields and algebras equipped with a derivation, which is a unary function satisfying the Leibniz product law...

. A differential algebra is an algebra with the extra operation of derivation (algebraic version of differentiation). Using the derivation operation new equations can be written and their solutions used in extensions

Field extension

In mathematics, more specifically in abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties.Field extensions can be...

 of the algebra. By starting with the field

Field (mathematics)

In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

 of rational function

Rational function

In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions.-Definitions:In the case of one variable, , a rational function is a function of the form...

s, two special types of transcendental extensions (the logarithm and the exponential) can be added to the field building a tower containing elementary functions.

A differential field F is a field F₀ (rational functions over the rationals

Rational number

In mathematics, a rational number is any number that can be expressed as the quotient a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer corresponds to a rational number. The set of all rational numbers is usually denoted .Formally each rational...

 Q for example) together with a derivation map u → ∂u. (Here ∂u is a new function. Sometimes the notation u ′ is used.) The derivation captures the properties of differentiation, so that for any two elements of the base field, the derivation is linear
and satisfies the Leibniz product rule

Product rule

In calculus, the product rule is a formula used to find the derivatives of products of functions.It may be stated thus:or in the Leibniz notation thus:.- Discovery by Leibniz :...

An element h is a constant if ∂h = 0. If the base field is over the rationals, care must be taken when extending the field to add the needed transcendental constants.

A function u of a differential extension F[u] of a differential field F is an elementary function over F if the function u

• is algebraic over F, or
• is an exponential, that is, ∂u = u ∂a for a ∈ F, or
• is a logarithm, that is, ∂u = ∂a / a for a ∈ F.

(this is Liouville's theorem).