Liouville's theorem (differential algebra)

In mathematics, Liouville's theorem, originally formulated by French mathematician Joseph Liouville in 1833 to 1841,[1][2][3] 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.

Definitions

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For any differential field   the constants of   is the subfield   Given two differential fields   and     is called a logarithmic extension 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 exponential extension 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 elementary differential extension of   if there is a finite chain of subfields from   to   where each extension in the chain is either algebraic, logarithmic, or exponential.

Basic theorem

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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.

A proof of Liouville's theorem can be found in section 12.4 of Geddes, et al.[4] See Lützen's scientific bibliography for a sketch of Liouville's original proof [5] (Chapter IX. Integration in Finite Terms), its modern exposition and algebraic treatment (ibid. §61).

Examples

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As an example, the field   of rational functions in a single variable has a derivation given by the standard derivative with respect to that variable. The constants of this field are just the complex numbers   that is,  

The function   which exists in   does not have an antiderivative in   Its antiderivatives   do, however, exist in the logarithmic extension  

Likewise, the function   does not have an antiderivative in   Its antiderivatives   do not seem to satisfy the requirements of the theorem, since they are not (apparently) sums of rational functions and logarithms of rational functions. However, a calculation with Euler's formula   shows that in fact the antiderivatives can be written in the required manner (as logarithms of rational functions).  

Relationship with differential Galois theory

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Liouville's theorem is sometimes presented as a theorem in differential Galois theory, but this is not strictly true. The theorem can be proved without any use of Galois theory. Furthermore, the Galois group of a simple antiderivative is either trivial (if no field extension is required to express it), or is simply the additive group of the constants (corresponding to the constant of integration). Thus, an antiderivative's differential Galois group does not encode enough information to determine if it can be expressed using elementary functions, the major condition of Liouville's theorem.

See also

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Notes

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  1. ^ Liouville 1833a.
  2. ^ Liouville 1833b.
  3. ^ Liouville 1833c.
  4. ^ Geddes, Czapor & Labahn 1992
  5. ^ Lützen, Jesper (1990). Joseph Liouville 1809–1882. Studies in the History of Mathematics and Physical Sciences. Vol. 15. New York, NY: Springer New York. doi:10.1007/978-1-4612-0989-8. ISBN 978-1-4612-6973-1.

References

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