Liouville's Theorem (differential Algebra)
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Liouville's theorem, originally formulated by French mathematician
Joseph Liouville Joseph Liouville ( ; ; 24 March 1809 – 8 September 1882) was a French mathematician and engineer. Life and work He was born in Saint-Omer in France on 24 March 1809. His parents were Claude-Joseph Liouville (an army officer) and Thérès ...
in 1833 to 1841, places an important restriction on
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a continuous function is a differentiable function whose derivative is equal to the original function . This can be stated ...
s that can be expressed as
elementary function In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, a ...
s. The antiderivatives of certain
elementary function In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, a ...
s cannot themselves be expressed as elementary functions. These are called nonelementary antiderivatives. A standard example of such a function is e^, whose antiderivative is (with a multiplier of a constant) the
error function In mathematics, the error function (also called the Gauss error function), often denoted by , is a function \mathrm: \mathbb \to \mathbb defined as: \operatorname z = \frac\int_0^z e^\,\mathrm dt. The integral here is a complex Contour integrat ...
, familiar from
statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
. Other examples include the functions \frac and x^x. Liouville's theorem states that elementary antiderivatives, if they exist, are in the same
differential field In mathematics, differential algebra is, broadly speaking, the area of mathematics consisting in the study of differential equations and differential operators as algebra, algebraic objects in view of deriving properties of differential equations ...
as the function, plus possibly a finite number of applications of the logarithm function.


Definitions

For any
differential field In mathematics, differential algebra is, broadly speaking, the area of mathematics consisting in the study of differential equations and differential operators as algebra, algebraic objects in view of deriving properties of differential equations ...
F, the of F is the subfield \operatorname(F) = \. Given two differential fields F and G, G is called a of F if G is a simple transcendental extension of F (that is, G = F(t) for some transcendental t) such that D t = \frac \quad \text s \in F. This has the form of a
logarithmic derivative In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function is defined by the formula \frac where is the derivative of . Intuitively, this is the infinitesimal relative change in ; that is, the in ...
. Intuitively, one may think of t as the
logarithm In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...
of some element s of F, in which case, this condition is analogous to the ordinary
chain rule In calculus, the chain rule is a formula that expresses the derivative of the Function composition, composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h ...
. However, F is not necessarily equipped with a unique logarithm; one might adjoin many "logarithm-like" extensions to F. Similarly, an is a simple transcendental extension that satisfies \frac = D s \quad \text s \in F. 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 of F if there is a finite chain of subfields from F to G where each
extension Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (proof theory) * Extension (predicate logic), the set of tuples of values that ...
in the chain is either algebraic, logarithmic, or exponential.


Basic theorem

Suppose F and G are differential fields with \operatorname(F) = \operatorname(G), and that G is an elementary differential extension of F. Suppose f \in F and g \in G satisfy D g = f (in words, suppose that G contains an antiderivative of f). Then there exist c_1, \ldots, c_n \in \operatorname(F) and f_1, \ldots, f_n, s \in F such that f = c_1 \frac + \dotsb + c_n \frac + D s. In other words, the only functions that have "elementary antiderivatives" (that is, antiderivatives living in, at worst, an elementary differential extension of F) 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. See Lützen's scientific bibliography for a sketch of Liouville's original proof (Chapter IX. Integration in Finite Terms), its modern exposition and algebraic treatment (ibid. §61).


Examples

As an example, the field F := \Complex(x) of
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
s in a single variable has a derivation given by the standard
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
with respect to that variable. The constants of this field are just the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s \Complex; that is, \operatorname(\Complex(x)) = \Complex, The function f := \tfrac, which exists in \Complex(x), does not have an antiderivative in \Complex(x). Its antiderivatives \ln x + C do, however, exist in the logarithmic extension \Complex(x, \ln x). Likewise, the function \tfrac does not have an antiderivative in \Complex(x). Its antiderivatives \tan^(x) + C 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 Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for ...
e^ = \cos \theta + i \sin \theta shows that in fact the antiderivatives can be written in the required manner (as logarithms of rational functions). \begin e^ & = \frac = \frac = \frac \\ \theta & = \frac \ln \left(\frac\right) \\ \tan^ x & = \frac \ln \left(\frac\right) \end


Relationship with differential Galois theory

Liouville's theorem is sometimes presented as a theorem in
differential Galois theory In mathematics, differential Galois theory is the field that studies extensions of differential fields. Whereas algebraic Galois theory studies extensions of field (mathematics), algebraic fields, differential Galois theory studies extensions of ...
, but this is not strictly true. The theorem can be proved without any use of
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...
. Furthermore, the
Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...
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


References

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External links

* {{MathWorld, id=LiouvillesPrinciple, title=Liouville's Principle Differential algebra Differential equations Field (mathematics) Theorems in algebra