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 see oxidation state."

In mathematics, an elementary function is a function built from a finite number of exponentials, logarithms, constants, one variable, and roots of equations through composition and combinations using the four elementary operations (+ – × ÷). The trigonometric functions 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 functions.

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 the implicit 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 by Joseph 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 of differential 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 of rational functions, 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.

Look at other dictionaries:

  • elementary function — elementarioji funkcija statusas T sritis fizika atitikmenys: angl. elementary function vok. elementare Funktion, f rus. элементарная функция, f pranc. fonction élémentaire, f …   Fizikos terminų žodynas

  • elementary function — noun Any function that is composed of algebraic functions, trigonometric functions, exponential functions and/or logarithmic functions, combined using addition, subtraction, multiplication and/or division …   Wiktionary

  • elementary function — Math. one of a class of functions that is generally taken to include power, exponential, and trigonometric functions, their inverses, and finite combinations of them. * * * …   Universalium

  • elementary function — Math. one of a class of functions that is generally taken to include power, exponential, and trigonometric functions, their inverses, and finite combinations of them …   Useful english dictionary

  • Elementary — Elementary: *Education ** Elementary education, consists of the first years of formal, structured education that occur during childhood. **Elementary school, a school providing elementary or primary education. Historically, a school in the UK… …   Wikipedia

  • ELEMENTARY — In computational complexity theory, the complexity class ELEMENTARY is the union of the classes in the exponential hierarchy.: egin{matrix} m{ELEMENTARY} = m{EXP}cup m{2EXP}cup m{3EXP}cupcdots = m{DTIME}(2^{n})cup m{DTIME}(2^{2^{n)cup… …   Wikipedia

  • Elementary class — In the branch of mathematical logic called model theory, an elementary class (or axiomatizable class) is a class consisting of all structures satisfying a fixed first order theory. Contents 1 Definition 2 Conflicting and alternative terminology …   Wikipedia

  • Function (mathematics) — f(x) redirects here. For the band, see f(x) (band). Graph of example function, In mathematics, a function associates one quantity, the a …   Wikipedia

  • Elementary substructure — In model theory, given two structures mathfrak A 0 and mathfrak A, both of a common signature Sigma, we say that mathfrak A 0 is an elementary substructure of mathfrak A (sometimes notated mathfrak A 0 preceq mathfrak A [Monk 1976: 331 (= Def. 19 …   Wikipedia

  • Elementary embedding — In model theory, an elementary embedding is a special case of an embedding that preserves all first order formulas.DefinitionGiven models M and N in the same language L, a function: f:M o Nis called an elementary embedding if f(M) is an… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”