Elementary function (differential algebra)
Encyclopedia
In mathematics
, an elementary function is a function
of one variable
built from a finite number of exponential
s, logarithm
s, constant
s, and nth roots through composition
and combinations using the four elementary operations
(+ – × ÷). By allowing these functions (and constants) to be complex numbers, trigonometric function
s and their inverses
become included in the elementary functions (see trigonometric functions and complex exponentials).
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 formulae for the roots (the formulae are elementary functions).
Elementary functions were introduced by Joseph Liouville
in a series of papers from 1833 to 1841. An algebraic treatment of elementary functions was started by Joseph Fels Ritt in the 1930s.
and
This last function is equal to the inverse cosine trigonometric function
in the entire complex domain. Hence, 
is an elementary function. An example of a function that is not elementary is the error function
a fact that cannot be seen directly from the definition of elementary function but can be proven using the Risch algorithm
.
. 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
of the algebra. By starting with the field
of rational function
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
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
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
(this is Liouville's theorem
).
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an elementary function is a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
of one variable
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...
built from a finite number of exponential
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...
s, logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...
s, constant
Coefficient
In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...
s, 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. It involves the study of quantity, especially as the result of combining numbers...
(+ – × ÷). 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 are the inverse functions of the trigonometric functions with suitably restricted domains .The notations sin−1, cos−1, etc...
become included in the elementary functions (see trigonometric functions and complex exponentials).
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 formulae for the roots (the formulae are elementary functions).
Elementary functions were introduced by Joseph Liouville
Joseph Liouville
- 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
-
.
This last function is equal to the inverse cosine trigonometric function
in the entire complex domain. Hence, is an elementary function. An example of a function that is not elementary is the error functionError function
In mathematics, the error function is a special function of sigmoid shape which occurs in probability, statistics 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 Henry 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 algebraDifferential algebra
In mathematics, differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with a derivation, which is a unary function that is linear and satisfies 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 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...
of the algebra. By starting with the field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it 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. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
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 or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
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 two or more functions. It may be stated thus:'=f'\cdot g+f\cdot g' \,\! or in the Leibniz notation thus:...
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 algebraicAlgebraic functionIn mathematics, an algebraic function is informally a function that satisfies a polynomial equation whose coefficients are themselves polynomials with rational coefficients. For example, an algebraic function in one variable x is a solution y for an equationwhere the coefficients ai are polynomial...
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
Liouville's theorem (differential algebra)
In mathematics, Liouville's theorem, originally formulated by Joseph Liouville in the 1830s and 1840s, places an important restriction on antiderivatives that can be expressed as elementary functions....
).