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This article discusses the concept of elementary functions in differential algebra. For simple functions see the list of mathematical functions.

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.

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:

and

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

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.

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 = u ∂a for a ∈ F, or
• is a logarithm, that is, ∂u = ∂a / a for a ∈ F.

(this is Liouville's theorem).

• Maxwell Rosenlicht (1972). "Integration in finite terms". American Mathematical Monthly 79: 963–972. 

• Joseph Fels Ritt, Differential Algebra, AMS, 1950.