In mathematics, an elementary function is a function built from a finite number of exponentials, logarithms, constants, one variable, and nth roots 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 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).
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:
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, too. 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.
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
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