Elementary function: Difference between revisions

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{{short description|A kind of mathematical function}}

{{distinguish|Elementary recursive function}}

In [[mathematics]], '''elementary functions''' are those [[function (mathematics)|function]]s that are most commonly encountered by beginners. They are typically [[real function]]s of a single real [[variable (mathematics)|variable]] that can be defined by applying the operations of [[addition]], [[multiplication]], [[division (mathematics)|division]], [[nth root|{{mvar|n}}th root]], [[function composition]], and [[inverse function|function inversioncomposition]] to [[Polynomial function|polynomial]], [[exponential function|exponential]], [[logarithm]], and [[Trigonometric functions|trigonometric]] functions. They include [[inverse trigonometric functions]], [[hyperbolic functions]] and [[inverse hyperbolic functions]], which can be expressed in terms of logarithms and exponential function.

All elementary functions have [[derivative]]s of any order, which are also elementary, and can be [[algorithmically]] computed by applying the [[differentiation rules]]. The [[Taylor series]] of an elementary function converges in a neighborhood of every point of its ___domain. More generally, they are [[global analytic function]]s, defined (possibly with multiple values, such as the elementary function <math>\sqrt z</math> or <math>\log z</math>) for every [[complex number|complex]] argument, except at [[isolated point]]s. In contrast, [[antiderivative]]s of elementary functions need not be elementary and is difficult to decide whether a specific elementary function has an elementary antiderivative.

In an attempt to solve this problem, [[Joseph Liouville]] introduced thein 1833 a notiondefinition of elementary functions inthat 1833extends the above one and is commonly accepted:<ref>{{harvnb|Liouville|1833a}}.</ref><ref>{{harvnb|Liouville|1833b}}.</ref><ref>{{harvnb|Liouville|1833c}}.</ref> An ''elementary function'' is a function that can be built, using addition, multiplication, division, and function composition, from [[constant function]]s, exponential functions, the [[complex logarithm]], and [[polynomial roots|roots]] of polynomials with elementary functions as coefficients. This includes the trigonometric functions, since, for example, {{tmath|1=\textstyle \cos x=\frac{e^{ix}+e^{-ix} }2}}, as well as every [[algebraic function]].

[[Liouville's theorem (differential algebra)|Liouville's result]] is that, if an elementary function has an elementary antiderivative, then this antiderivative is a linear combination of logarithms, where the coefficients and the arguments of the logarithms are elementary functions involved, in some sense, in the definition of the function. More than 130 years later, [[Risch algorithm]], named after [[Robert Henry Risch]], is an algorithm to decide whether an elementary function has an elementary antiderivative, and, if it has, to compute this antiderivative. Despite dealing with elementary functions, the Risch algorithm is far from elementary; {{as of|2025|lc=y}}, it seems that no complete implementation is available.

== Examples ==

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=== Basic examples ===

Elementary functions of a single variable {{mvar|x}} include:

* [[Constant function]]s: <math>2,\ \pi,\ e,</math>, the [[Euler–Mascheroni constant]], [[Apéry's constant]], [[Khinchin's constant]], etc. Any constant real (or complex) number.

* [[Exponentiation|Powers of {{mvartmath|x}}]]: <math>x,\ x^2,\ \sqrt{x}\ (xalpha=e^\frac{1}{2}),\ x^\frac{2}{3},x^\pi,alpha\log x^e, x^{\sqrt{-1}},</math> etc. (The exponent can be any real or complex constant.)

* [[Exponential function]]s: <math>\textstyle e^x, \quad a^x=e^{x\log a}</math>

* [[Logarithm]]s: <math>\textstyle \log x, \ quad\log_a x=\frac {\log x}{\log a}</math>

* [[Trigonometric function]]s: <math>\textstyle\sin x=\frac{e^{ix}-e^{-ix}}{2i},\ \cos x=\frac{e^{ix}+e^{-ix}}{2},\ \tan x=\frac{\sin x}{\cos x},\ </math> etc.

* [[Inverse trigonometric function]]s: <math>\arcsin x,\ \arccos x,</math> etc.

* [[Hyperbolic function]]s: <math>\sinh x,\ \cosh x,</math> etc.

* [[Inverse hyperbolic function]]s: <math>\operatorname{arsinh} x,\ \operatorname{arcosh} x,</math> etc.

* All functions obtained by adding, subtracting, multiplying or dividing a finite number of any of the previous functions<ref>{{cite book|author=Morris Tenenbaum|title=Ordinary Differential Equations|date=1985|publisher=Dover|isbn=0-486-64940-7|page=[https://archive.org/details/ordinarydifferen00tene_0/page/17 17]|url-access=registration|url=https://archive.org/details/ordinarydifferen00tene_0/page/17}}</ref>

* All functions obtained byas [[root extractionof a polynomial|roots]] of a polynomial withwhose coefficients inare elementary functions<ref name=":1">{{Cite book|title=Calculus|last=Spivak, Michael.|date=1994|publisher=Publish or Perish|isbn=0914098896|edition=3rd|___location=Houston, Tex.|pages=363|oclc=31441929}}</ref><ref>Ritt, chapter 1</ref>

* All functions obtained by [[function composition|composing]] a finite number of any of the previously listed functions

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=== Non-elementary functions ===

All elementary functions are [[Analytic function|analytic]], unlikein the following sense: they can be extended to [[Absolutefunctions value|absoluteof valuea complex variable]] (possibly [[multivalued function|multivalued]]) orthat discontinuousare functionsanalytic suchexcept asat finitely many points of the [[stepcomplex functionplane]].<ref>{{Cite journal |last=Risch |first=Robert H. |date=1979 |title=Algebraic Properties of the Elementary Functions of Analysis |url=https://www.jstor.org/stable/2373917 |journal=[[American Journal of Mathematics]] |volume=101 |issue=4 |pages=743–759 |doi=10.2307/2373917 |jstor=2373917 |issn=0002-9327|url-access=subscription }}</ref> Thus nonanalytic functions such as the [[absolute value]] function are not elementary,<ref>Watson and Whittaker 1927, footnote to p 82</ref> Some have proposed. extendingIn the setcontext toof include,elementary for examplefunctions, the [[Lambertfunction W function]]<refmath>{{Cite journal |lasty=Stewart |first=Seán |date=2005 |title=A new elementary function for our curricula? |url=https:f(x)<//files.eric.ed.gov/fulltext/EJ720055.pdfmath> |journal=Australiandefined Senioras Mathematicsthe Journalroot |volume=19of |issue=<math>y^2-x^2 |pages=8–26}}0</refmath> oris [[elliptictwo-valued: function]]s,<refmath>Ince,y=\pm footnote to p 330x</math>.</ref> allnor ofare whichmost areother still[[piecewise-defined analyticfunction]]s.

Not every analytic function is elementary. SomeIn examplesfact, thatmost [[special function]]s are ''not'' elementary,. underNon-elementary standardfunctions definitionsinclude:

* [[tetration]]

* the [[gamma function]]

* non-elementary [[Liouvillian function#Examples|Liouvillian functions]], including

** the [[exponential integral]] (''Ei''), [[logarithmic integral]] (''Li'' or ''li'') and [[Fresnel integral|Fresnel integrals]] (''S'' and ''C'').

** the [[error function]], <math>\mathrm{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\,dt,</math> a fact that may not be immediately obvious, but can be proven using the [[Risch algorithm]].

* other [[nonelementary integral]]s, including the [[Dirichlet integral]] and [[elliptic integral]].

== Closure ==

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==Differential algebra==

Some have proposed extending the set of elementary functions by extending with certain [[transcendental function]]s, to include, for example, the [[Lambert W function]]<ref>{{Cite journal |last=Stewart |first=Seán |date=2005 |title=A new elementary function for our curricula? |url=https://files.eric.ed.gov/fulltext/EJ720055.pdf |journal=Australian Senior Mathematics Journal |volume=19 |issue=2 |pages=8–26}}</ref> or [[elliptic function]]s,<ref>Ince, E. L. (1956) [1926]. ''Ordinary Differential Equations''. New York: Dover Publications. ISBN 0-486-60339-4, footnote to p 330</ref> all of which are analytic. The key attribute, from the perspective of the Liouville theorem, is that as a class, they are closed under taking derivatives. For example, the Lambert function <math>w=W(z)</math>, which is defined implicitly by the equation <math>we^w=z</math>, has a derivative which can be obtained by [[implicit differentiation]]:

<math>W'(z) = \frac{e^{-W(z)}}{1+W(z)},</math>

which is again "elementary", provided that <math>W(z)</math> is.

The mathematical definition of an '''elementary function''', or a function in elementary form, is consideredformalized in the context of [[differential algebra]]. A [[differential algebrafield]] is ana [[field (mathematics)|field]] algebra with thean extra operation of derivation (algebraic version of differentiation). Using the derivation operation new equations can be written and their solutions used in [[field extension|extensions]] of the algebra. By starting with the [[field (mathematics)|field]] of [[rational function]]s, two special types of transcendental extensions (the logarithm and the exponential) can be added to the field building a tower containing elementary functions.

A '''differential field''' ''F'' is a field ''F''₀ (rational functions over the [[rational number|rationals]] '''Q''' for example) together with a derivation map ''u''&nbsp;→&nbsp;∂''u''. (Here ∂''u'' is a new function. Sometimes the notation ''u''&prime; is used.) The derivation captures the properties of differentiation, so that for any two elements of the base field, the derivation is linear

: <math>\partial (u + v) = \partial u + \partial v </math>

A ''differential field'' {{tmath|F}} is a field together with a [[derivation (differential algebra)|derivation]] {{tmath|u\mapsto \partial u}} that maps {{tmath|F}} to itself. The derivation generalizes [[derivative]], being linear (thaat is, {{tmath|1=\partial (u + v) = \partial u + \partial v}}) and satisfying the [[product rule|Leibniz product rule]] (that is,{{tmath|1=\partial(u\cdot v)=\partial u\cdot v+u\cdot\partial v}}) for every two elements {{tmath|u}} and {{tmath|v}} in {{tmath|F}}. The [[rational function]]s over {{tmath|\Q}} of {{tmath|\C}} form a basic examples of differential fields, when equipped with the usual derivative.

and satisfies the [[product rule|Leibniz product rule]]

An element {{math|''h''}} of {{tmath|F}} is a constant if {{tmath|1=\partial h=0}}. The constants of {{tmath|F}} form a dfferential field with zero derivative. Care must be taken that a differential field extension of a differential field may enlarge the field of constants.

: <math>\partial(u\cdot v)=\partial u\cdot v+u\cdot\partial v\,.</math>

A function {{mvar|u}} of a differential extension {{mvar|G}} of a differential field {{mvar|F}} is an '''elementary function''' over {{mvar|F}} if it belongs to a finite chain (for inclusion) of differential subfields of {{mvar|G}} that starts from {{mvar|F}} and is such that each is generated over the preceding one by a function that is either

An element ''h'' is a constant if ''∂h&nbsp;=&nbsp;0''. If the base field is over the rationals, care must be taken when extending the field to add the needed transcendental constants.

* is [[Algebraic function|algebraic]] over ''F''the preceding field, or

* is an '''exponential''', that is, ∂''{{tmath|1=\partial u'' = ''u''\partial ∂''a''}} for ''some {{tmath|a''\in ∈ ''F''}}, or

* a ''logarithm'', that is, {{tmath|1=\partial u = \partial a/a}} for some {{tmath|a\in F}}.

(see also [[Liouville's theorem (differential algebra)|Liouville's theorem]])

With this definition, the usual elementary functions are exactly the function that are elementary over the field of the [[rational function]]s. This generalized definition allows considering every transcendental function as elementary for applying Liouville's theorem.

A function ''u'' of a differential extension ''F''[''u''] of a differential field ''F'' is an '''elementary function''' over ''F'' if the function ''u''

* is [[Algebraic function|algebraic]] over ''F'', or

* is an '''exponential''', that is, ∂''u'' = ''u'' ∂''a'' for ''a'' ∈ ''F'', or

* is a '''logarithm''', that is, ∂''u'' = ∂''a''&nbsp;/&nbsp;a for ''a'' ∈ ''F''.

(see also [[Liouville's theorem (differential algebra)|Liouville's theorem]])

==See also==