Liouville's Theorem (differential Algebra)

Liouville's Theorem (differential Algebra)

In mathematics, Liouville's theorem, originally formulated by Joseph Liouville in the 1830s and 1840s, places an important restriction on antiderivatives that can be expressed as elementary functions.

The antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions. A standard example of such a function is whose antiderivative is (with a multiplier of a constant) the error function, familiar from statistics. Other examples include the functions and

Liouville's theorem states that elementary antiderivatives, if they exist, must be in the same differential field as the function, plus possibly a finite number of logarithms.