- Elementary function
: "This article discusses the concept of elementary functions in differential algebra. For simple functions see the

list of mathematical functions . For the concept of elementary form of an atom seeoxidation state ."In

mathematics , an**elementary function**is a function built from a finite number of exponentials,logarithm s,constant s, onevariable , and roots of equations through composition and combinations using the four elementary operations (+ – × ÷). Thetrigonometric function s and their inverses are assumed to be included in the elementary functions by using complex variables and the relations between the trigonometric functions and the exponential and logarithm functions.Elementary functions are considered a subset of

special function s.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 and theimplicit function theorem assures the existence of a function that returns each one of the roots of a polynomial equation.Examples of elementary functions include:

: $frac\{e^\{\; an(x)\{1-x^2\}sinleft(sqrt\{1+ln^2\; x\},\; ight)$

and

: $,ln(-x^2).$

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 : $mathrm\{erf\}(x)=frac\{2\}\{sqrt\{piint\_0^x\; e^\{-t^2\},dt,$

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

Risch algorithm .Elementary functions were introduced by

Joseph Liouville in a series of papers from 1833 to 1841. An algebraic treatment of elementary functions was started byJoseph Fels Ritt in the 1930s.**Differential algebra**The mathematical definition of an

**elementary function**, or a function in elementary form, is considered in the context ofdifferential algebra . 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 ofrational 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 "F"_{0}(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: $partial\; (u\; +\; v)\; =\; partial\; u\; +\; partial\; v$

and satisfies the Leibniz' product rule

: $partial(ucdot\; v)=partial\; ucdot\; v+ucdotpartial\; v,.$

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).**References***

*Joseph Ritt , " [*http://www.ams.org/online_bks/coll33/ Differential Algebra*] ", AMS, 1950.

*Wikimedia Foundation.
2010.*