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Elementary function (differential algebra)

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Elementary function (differential algebra)

For the complexity class, see ELEMENTARY. For the logical system, see Elementary function arithmetic.

In mathematics, an elementary function is a function of one variable built from a finite number of exponentials, logarithms, constants, and roots of polynomial equations through composition and combinations using the four elementary operations (+ – × ÷). By allowing these functions (and constants) to be complex numbers, trigonometric functions and their inverses become included in the elementary functions (see trigonometric functions and complex exponentials).

The roots of polynomial 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.

Examples

Examples of elementary functions include:

Addition e.g. (x+1)
Multiplication e.g. (2x)
\frac{e^{\tan(x)}}{1+x^2}\sin\left(\sqrt{1+\ln^2 x}\,\right)

and

-i\ln(x+i\sqrt{1-x^2}), \,

This last function is equal to the inverse cosine trigonometric function \arccos(x) in the entire complex domain. Hence, \arccos(x) is an elementary function. An example of a function that is not elementary is the error function

\mathrm{erf}(x)=\frac{2}{\sqrt{\pi}}\int_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.

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 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

\partial (u + v) = \partial u + \partial v

and satisfies the Leibniz product rule

\partial(u\cdot v)=\partial u\cdot v+u\cdot\partial 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 = ua for aF, or
  • is a logarithm, that is, ∂u = ∂a / a for aF.

(this is Liouville's theorem).

See also

References

  • Differential Algebra, AMS, 1950.

External links

  • at Encyclopaedia of Mathematics
  • MathWorld.
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