Elementary function

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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 nth roots through composition and combinations using the four elementary operations (+ – × ÷). By considering these functions (and constants) to be complex, the elementary function concept is enlarged to include trigonometric functions and their inverses (see trigonometric functions and complex exponentials).

The roots of a polynomial equation are the solutions of a polynomial equation with constant coefficients. For polynomials of degree four and less there are explicit formulae for determining the roots (the formulae are elementary functions), but elementary functions are not sufficient for finding the roots of general higher-degree polynomials.

Note that some elementary functions, such as roots, logarithms, or inverse trigonometric functions, are not entire functions and their definition may be ambiguous, especially for non-real numbers.

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 of elementary functions include:

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


  • -i\ln(x+i\sqrt{1-x^2})

The 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 may not be immediately obvious, but can be proven using the Risch algorithm.

Differential algebra[edit]

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).

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