Liouville's theorem (differential algebra)

In mathematics, differential Galois theory is a theory based on the model of Galois theory which studies which functions have 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 $e^{-x^{2}}$, whose antiderivative is (up to constants) the error function, familiar from statistics. Other examples include the functions

${\frac {\sin(x)}{x}}$

and

$x^{x}.\,$

The machinery of differential Galois theory allows one to determine when an elementary function does or does not have an antiderivative that can be expressed as an elementary function. Whereas algebraic Galois theory studies extensions of algebraic fields, differential Galois theory studies extensions of differential fields, i.e. fields that are equipped with a derivation, D. Much of the theory of differential Galois theory is parallel to algebraic Galois theory. One difference between the two constructions is that the Galois groups in differential Galois theory tend to be matrix Lie groups, as compared with the finite groups often encountered in algebraic Galois theory. The problem of finding which integrals of elementary functions can be expressed with other elementary functions is analogous to the problem of solutions of polynomial equations by radicals in algebraic Galois theory, and is solved by Picard–Vessiot theory.

Definitions

For any differential field F, there is a subfield

Con(F) = {f in F | Df = 0},

called the constants of F. Given two differential fields F and G, G is called a logarithmic extension of F if G is a simple transcendental extension of F (i.e. G = F(t) for some transcendental t) such that

Dt = Ds/s for some s in F.

This has the form of a logarithmic derivative. Intuitively, one may think of t as the logarithm of some element s of F, in which case, this condition is analogous to the ordinary chain rule. But it must be remembered that F is not necessarily equipped with a unique logarithm; one might adjoin many "logarithm-like" extensions to F. Similarly, an exponential extension is a simple transcendental extension which satisfies

Dt = tDs.

With the above caveat in mind, this element may be thought of as an exponential of an element s of F. Finally, G is called an elementary differential extension of F if there is a finite chain of subfields from F to G where each extension in the chain is either algebraic, logarithmic, or exponential.

Examples of defined terms

As an example, the field C(x) 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 C.

Basic theorem

The basic theorem of differential Galois theory, originally due to Joseph Liouville in the 1830s and 1840s and hence referred to as Liouville's theorem, is as follows.

Suppose F and G are differential fields, with Con(F) = Con(G), and that G is an elementary differential extension of F. Let a be in F, y in G, and suppose Dy = a (in words, suppose that G contains an antiderivative of a). Then there exist c₁, ..., c_n in Con(F), u₁, ..., u_n, v in F such that

$a=c_{1}{\frac {Du_{1}}{u_{1}}}+\dotsb +c_{n}{\frac {Du_{n}}{u_{n}}}+Dv.$

In other words, the only functions that have "elementary antiderivatives" (i.e. antiderivatives living in, at worst, an elementary differential extension of F) are those with this form prescribed by the theorem. 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.

References

• Bertrand, D. (1996), "Review of "Lectures on differential Galois theory"" (PDF), Bulletin of the American Mathematical Society, 33 (2), ISSN 0002-9904
• Magid, Andy R. (1994), Lectures on differential Galois theory, University Lecture Series, vol. 7, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-7004-4, MR1301076
• Magid, Andy R. (1999), "Differential Galois theory" (PDF), Notices of the American Mathematical Society, 46 (9): 1041–1049, ISSN 0002-9920, MR1710665
• van der Put, Marius; Singer, Michael F. (2003), Galois theory of linear differential equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 328, Berlin, New York: Springer-Verlag, ISBN 978-3-540-44228-8, MR1960772

See also

• Risch algorithm