Elementary function

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).

The elementary functions include the trigonometric and hyperbolic functions and their inverses, as they are expressible with complex exponentials and logarithms.

It follows directly from the definition that the set of elementary functions is closed under arithmetic operations and composition. It is also closed under differentiation. It is 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.

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

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.

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