Liouville's theorem (differential algebra)

In mathematics, Liouville's theorem, originally formulated by French mathematician Joseph Liouville in 1833 to 1841, 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. These are called nonelementary antiderivatives. A standard example of such a function is ${\displaystyle e^{-x^{2}},}$ whose antiderivative is (with a multiplier of a constant) the error function, familiar in statistics. Other examples include the functions ${\displaystyle {\frac {\sin(x)}{x}}}$ and ${\displaystyle x^{x}.}$

Liouville's theorem states that if an elementary function has an elementary antiderivative, then the antiderivative can be expressed only using logarithms and functions that are involved, in some sense, in the original elementary function. An example is the antiderivative of ${\displaystyle \sec x}$ is ${\displaystyle \log |\sec x+\tan x|}$, which uses only logarithms and trigonometric functions. More precisely, Liouville's theorem states that elementary antiderivatives, if they exist, are in the same differential field as the function, plus possibly a finite number of applications of the logarithm function.

The Liouville theorem is a precursor to the Risch algorithm, which relies on the Liouville theorem to find any elementary antiderivative.