In mathematics, the Liouvillian functions comprise a set of functions including the

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 and their repeated

integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...

s. Liouvillian functions can be recursively defined as integrals of other Liouvillian functions. More explicitly, a Liouvillian function is a function of one variable which is the

composition Composition or Compositions may refer to: Arts and literature * Composition (dance), practice and teaching of choreography *Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include ...

of a finite number of

arithmetic operations Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...

exponential Exponential may refer to any of several mathematical topics related to exponentiation, including: *Exponential function, also: **Matrix exponential, the matrix analogue to the above *Exponential decay, decrease at a rate proportional to value * Exp ...

s, constants, solutions of algebraic equations (a generalization of ''n''th roots), and

antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...

s. The

logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...

function does not need to be explicitly included since it is the integral of

1/x

. It follows directly from the definition that the set of Liouvillian functions is closed under arithmetic operations, composition, and integration. It is also closed under differentiation. It is not closed under

limits Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...

and infinite sums. Liouvillian functions were introduced by

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 Thérèse ...

in a series of papers from 1833 to 1841.

Examples

All

s are Liouvillian. Examples of well-known functions which are Liouvillian but not elementary are the nonelementary antiderivatives, for example: * The

error function In mathematics, the error function (also called the Gauss error function), often denoted by , is a complex function of a complex variable defined as: :\operatorname z = \frac\int_0^z e^\,\mathrm dt. This integral is a special (non- elementa ...

\mathrm(x)=\frac\int_0^x e^\,dt,

* The

(''Ei''),

logarithmic Logarithmic can refer to: * Logarithm, a transcendental function in mathematics * Logarithmic scale, the use of the logarithmic function to describe measurements * Logarithmic spiral, * Logarithmic growth * Logarithmic distribution, a discrete pr ...

(''Li'' or ''li'') and Fresnel (''S'' and ''C'') integrals. All Liouvillian functions are solutions of

algebraic differential equation In mathematics, an algebraic differential equation is a differential equation that can be expressed by means of differential algebra. There are several such notions, according to the concept of differential algebra used. The intention is to i ...

s, but not conversely. Examples of functions which are solutions of algebraic differential equations but not Liouvillian include:L. Chan, E.S. Cheb-Terrab, "Non-liouvillian solutions for second order Linear ODEs", ''Proceedings of the 2004 international symposium on Symbolic and algebraic computation (ISSAC '04)'', 2004, pp. 80–86 * the

Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...

s (except special cases); * the hypergeometric functions (except special cases). Examples of functions which are ''not'' solutions of algebraic differential equations and thus not Liouvillian include all transcendentally transcendental functions, such as: * the

gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except th ...

; * the zeta function.

Examples

See also

References

Further reading