View source for 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 [[antiderivative]]s that can be expressed as elementary functions.

The antiderivatives of certain [[elementary function]]s cannot themselves be expressed as elementary functions.  A standard example of such a function is <math> e^{ - x^2 } </math>, whose antiderivative is (up to constants) the [[error function]], familiar from [[statistics]].  Other examples include the functions

:<math> \frac{ \sin ( x ) }{ x } </math>

and

:<math>x^x.\,</math>

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.

== Definitions ==

For any differential field ''F'', there is a subfield 

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

called the [[constant]]s of ''F''. Given two differential fields ''F'' and ''G'', ''G'' is called a '''logarithmic extension''' of ''F'' if ''G'' is a [[field extension|simple transcendental extension]] of ''F'' (i.e. ''G'' = ''F''(''t'') for some [[transcendence (mathematics)|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.

== Basic theorem ==

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 ''c''<sub>1</sub>, ..., ''c''<sub>n</sub> in Con(''F''), ''u''<sub>1</sub>, ..., ''u''<sub>''n''</sub>, ''v'' in ''F'' such that

:<math>a = c_1\frac{Du_1}{u_1}+\dotsb+c_n\frac{Du_n}{u_n}+Dv.</math>

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.

== Examples ==

As an example, the field '''C'''(''x'') of [[rational function]]s 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 number]]s '''C'''.

The function <math> \frac{1}{x} </math>, which exists in '''C'''(''x''), does not have an antiderivative in '''C'''(''x'').  Its antiderivatives <math>\ln x + C</math> do, however, exist in the logarithmic extension '''C'''(''x'', ln x).

Likewise, the function <math>\frac{1}{x^2+1}</math> does not have an antiderivative in '''C'''(''x'').  Its antiderivatives
<math>\tan^{-1} x + C</math> 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).

:<math>
\begin{align}
e^{i \theta}  & = \cos \theta + i \sin \theta \\
e^{-i \theta} & = \cos \theta - i \sin \theta \\
e^{2i \theta} & = \frac{e^{i \theta}}{e^{-i \theta}} = \frac{\cos \theta + i \sin \theta}{\cos \theta - i \sin \theta} \\
              & = \frac{1 + i \tan \theta}{1 - i \tan \theta} \\
2i \theta & = \ln \frac{1 + i \tan \theta}{1 - i \tan \theta} \\
2i \tan^{-1} x & = \ln \frac{1 + ix}{1 - ix} \\
\tan^{-1} x & = - \frac{1}{2} i \ln \frac{1+ix}{1-ix}
\end{align}
</math>

== Relationship with Differential Galois Theory ==

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 else 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.
<!--
== Example of theorem ==
Suppose we want to know whether a function of the form f*e<sup>g</sup> has an elementary antiderivative, with ''f'' and ''g'' in ''C''(''x'') -->

==References==
*{{Citation | last1=Bertrand | first1=D. | title=Review of "Lectures on differential Galois theory" | url=http://www.ams.org/bull/1996-33-02/S0273-0979-96-00652-0/S0273-0979-96-00652-0.pdf | year=1996 | journal=[[Bulletin of the American Mathematical Society]] | issn=0002-9904 | volume=33 | issue=2}}
*{{Citation | last1=Magid | first1=Andy R. | title=Lectures on differential Galois theory | url=http://books.google.com/books?id=cJ9vByhPqQ8C | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=University Lecture Series | isbn=978-0-8218-7004-4 | id={{MR|1301076}} | year=1994 | volume=7}}
*{{Citation | last1=Magid | first1=Andy R. | title=Differential Galois theory | url=http://www.ams.org/notices/199909/fea-magid.pdf | id={{MR|1710665}} | year=1999 | journal=[[Notices of the American Mathematical Society]] | issn=0002-9920 | volume=46 | issue=9 | pages=1041–1049}}
*{{Citation | last1=van der Put | first1=Marius | last2=Singer | first2=Michael F. | title=Galois theory of linear differential equations | url=http://www4.ncsu.edu/~singer/ms_papers.html | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] | isbn=978-3-540-44228-8 | id={{MR|1960772}} | year=2003 | volume=328}}

== See also ==
*[[Risch algorithm]]

[[Category:Field theory]]
[[Category:Differential algebra]]
[[Category:Differential equations]]