More explicitly, a Liouvillian function is a function of one variable which is the composition of a finite number of arithmetic operations (+, −, ×, ÷), exponentials, constants, solutions of algebraic equations (a generalization of nth roots), and antiderivatives.
The logarithm function does not need to be explicitly included since it is the integral of
It follows directly from the definition that the set of Liouvillian functions is closed under arithmetic operations, composition, and integration.
[example needed] Liouvillian functions were introduced by Joseph Liouville in a series of papers from 1833 to 1841.
Examples of well-known functions which are Liouvillian but not elementary are the nonelementary antiderivatives, for example: All Liouvillian functions are solutions of algebraic differential equations, but not conversely.