In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponential functions, and their inverses (e.g., arcsin, log, or x1/n).
Elementary functions were introduced by Joseph Liouville in a series of papers from 1833 to 1841.
[2][3][4] An algebraic treatment of elementary functions was started by Joseph Fels Ritt in the 1930s.
[5] Many textbooks and dictionaries do not give a precise definition of the elementary functions, and mathematicians differ on it.
Additionally, certain classes of functions may be obtained by others using the final two rules.
composed with addition, subtraction, and division provides the hyperbolic functions, while initial composition with
Some have proposed extending the set to include, for example, the Lambert W function.
[10] Some examples of functions that are not elementary: It follows directly from the definition that the set of elementary functions is closed under arithmetic operations, root extraction and composition.
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.
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.