Liouville's theorem (differential algebra)

In mathematics, Liouville's theorem, originally formulated by 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 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, are in the same differential field as the function, plus possibly a finite number of applications of the logarithm function. For any differential field the of is the subfield Given two differential fields and is called a of if is a simple transcendental extension of (that is, for some transcendental ) such that This has the form of a logarithmic derivative. Intuitively, one may think of as the logarithm of some element of in which case, this condition is analogous to the ordinary chain rule. However, is not necessarily equipped with a unique logarithm; one might adjoin many "logarithm-like" extensions to Similarly, an is a simple transcendental extension that satisfies With the above caveat in mind, this element may be thought of as an exponential of an element of Finally, is called an of if there is a finite chain of subfields from to where each extension in the chain is either algebraic, logarithmic, or exponential. Suppose and are differential fields with and that is an elementary differential extension of Suppose and satisfy (in words, suppose that contains an antiderivative of ). Then there exist and such that In other words, the only functions that have "elementary antiderivatives" (that is, antiderivatives living in, at worst, an elementary differential extension of ) are those with this form. Thus, on an intuitive level, the theorem states that the only elementary antiderivatives are the "simple" functions plus a finite number of logarithms of "simple" functions.