Liouvillian function

Article summary

In mathematics, the Liouvillian functions comprise a set of functions including the elementary functions and their repeated integrals. Liouvillian functions can be recursively defined as integrals of other Liouvillian functions. More explicitly, a Liouvillian function is a function of one variable which is the composition of a finite number of algebraic operations ( addition, subtraction, multiplication, division, exponentiation, and root extraction) and antiderivatives. The logarithmic function does not need to be explicitly included since it is the integral of 1/x . It follows directly from the definition that the set of Liouvillian functions is closed under arithmetic operations, composition, and integration. It is also closed under differentiation. It is not closed under limits and infinite sums. Liouvillian functions were introduced by Joseph Liouville in a series of papers from 1833 to 1841.

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