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In mathematics, an elementary function is a function of one variable which is the composition of a finite number of arithmetic operations (+ – × ÷), exponentials, logarithms, constants, and solutions of algebraic equations (a generalization of nth roots).

Elementary functions were introduced by Joseph Liouville in a series of papers from 1833 to 1841.[1][2][3] An algebraic treatment of elementary functions was started by Joseph Fels Ritt in the 1930s.[4]



Basic examplesEdit

The elementary functions (of x) include:

  • Powers of   etc.
  • Roots of   etc.
  • Exponential functions:  
  • Logarithms:  
  • Trigonometric functions:   etc.
  • Inverse trigonometric functions:   etc.
  • Hyperbolic functions:   etc.
  • All functions obtained by replacing x with any of the previous functions
  • All functions obtained by adding, subtracting, multiplying or dividing any of the previous functions[5]

Some elementary functions, such as roots, logarithms, or inverse trigonometric functions, are not entire functions and may be multivalued.

It follows that the absolute value function is also elementary as it can be expressed as  .

Composite examplesEdit

Examples of elementary functions include:

  • Addition, e.g. (x+1)
  • Multiplication, e.g. (2x)
  • Polynomial functions

The last function is equal to  , the inverse cosine, in the entire complex plane. Hence, it is an elementary function.

Non-elementary functionsEdit

An example of a function that is not elementary is the error function


a fact that may not be immediately obvious, but can be proven using the Risch algorithm.


It follows directly from the definition that the set of elementary functions is closed under arithmetic operations and composition. The elementary functions are closed under differentiation.[citation needed] They are not closed under limits and infinite sums. Importantly, the elementary functions are not closed under integration, as shown by Liouville's theorem, see Nonelementary integral. The Liouvillian functions are defined as the elementary functions and, recursively, the integrals of the Liouvillian functions.

Differential algebraEdit

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


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

  • 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 alsoEdit


  1. ^ Liouville 1833a.
  2. ^ Liouville 1833b.
  3. ^ Liouville 1833c.
  4. ^ Ritt 1950.
  5. ^ Ordinary Differential Equations. Dover. 1985. p. 17. ISBN 0-486-64940-7.


Further readingEdit

  • Davenport, J. H.: What Might "Understand a Function" Mean. In: Kauers, M.; Kerber, M., Miner, R.; Windsteiger, W.: Towards Mechanized Mathematical Assistants. Springer, Berlin/Heidelberg 2007, p. 55-65. [1]

External linksEdit