View source for Elementary function

{{Short description|Type of mathematical function}}
{{distinguish|Elementary recursive function}}
In [[mathematics]], an '''elementary function''' is a [[function (mathematics)|function]] of a single [[variable (mathematics)|variable]] ([[Function of a real variable|real]] or [[Complex analysis#Complex functions|complex]]) that is typically encountered by beginners.  The basic elementary functions are [[polynomial function]]s, [[rational function]]s, the [[trigonometric function]]s, the [[exponential function|exponential]] and [[logarithm]] functions, the [[n-th root]], and the [[inverse trigonometric function]]s, as well as those functions obtained by [[addition]], [[multiplication]], [[division (mathematics)|division]], and [[function composition|composition]] of these.  Some functions which are encountered by beginners are ''not'' elementary, such as the [[absolute value]] function and [[piecewise-defined function]]s. More generally, in modern mathematics, elementary functions comprise the set of functions previously enumerated, all [[algebraic function]]s (not often encountered by beginners), and all functions obtained by [[roots of a polynomial]] whose coefficients are elementary. 

This list of elementary functions was originally set forth by [[Joseph Liouville]] in 1833. A key property is that all elementary functions have [[derivative]]s of any order, which are also elementary, and can be [[algorithmically]] computed by applying the [[differentiation rules]] (or the rules for [[implicit differentiation]] in the case of roots). 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 [[multivalued function|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.

[[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.

In late-nineteenth-century analysis, elementary functions were often classified into successive kinds according to the number of independent integrations required for their definition.  Functions expressible without any integration—those generated from rational functions by algebraic operations together with exponentiation, logarithms, and circular or hyperbolic trigonometric functions—were said to be elementary functions of the first kind (in the sense of Liouville).  Functions defined by a single integration of an algebraic function, such as the error function and the elliptic integrals, were elementary functions of the second kind; their inverses, the elliptic functions, were considered of the same order.  Higher "kinds" (third, fourth, etc.) corresponded to multiple integrals of algebraic functions, giving rise to hyperelliptic and more general Abelian functions.{{sfn|Forsyth|1893}}

The essential point of the classification was that the class of elementary functions of any given kind be closed under the elementary operations—addition, multiplication, composition, and differentiation—so that differentiation never leads outside the same class, while integration may ascend to the next higher kind.

== Examples ==

=== 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 {{tmath|x}}]]: <math>x^\alpha=e^{\alpha\log x}</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 as [[root of a polynomial|roots]] of a polynomial whose coefficients are 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

Certain elementary functions of a single complex variable {{mvar|z}}, such as <math>\sqrt{z}</math> and <math>\log z</math>, may be [[multivalued function|multivalued]]. Additionally, certain classes of functions may be obtained by others using the final two rules. For example, the exponential function <math>e^{z}</math> composed with addition, subtraction, and division provides the hyperbolic functions, while initial composition with <math>iz</math> instead provides the trigonometric functions.

=== Composite examples ===
Examples of elementary functions include:

* Addition, e.g. ({{mvar|x}} + 1)
* Multiplication, e.g. (2{{mvar|x}})
*[[Polynomial]] functions
*<math>\frac{e^{\tan x}}{1+x^2}\sin\left(\sqrt{1+(\log x)^2}\right)</math>
*<math>-i\log\left(x+i\sqrt{1-x^2}\right) </math>

The last function is equal to <math>\arccos x</math>, the [[Inverse_trigonometric_functions#Logarithmic_forms|inverse cosine]], in the entire [[complex plane]].

All [[monomial]]s, [[polynomial]]s, [[rational function]]s and [[algebraic function]]s are elementary.

=== Non-elementary functions ===
All elementary functions are [[Analytic function|analytic]] in the following sense: they can be extended to [[functions of a complex variable]] (possibly [[multivalued function|multivalued]]) that are analytic except at finitely many points of the [[complex plane]].<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. In the context of elementary functions, the function <math>y=f(x)</math> defined as the root of <math>y^2-x^2=0</math> is two-valued: <math>y=\pm x</math>.</ref> nor are most other [[piecewise-defined function]]s.

Not every analytic function is elementary. In fact, most [[special function]]s are not elementary. Non-elementary functions include:
* 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 ==
It follows directly from the definition that the set of elementary functions is [[closure (mathematics)|closed]] under arithmetic operations, (algebraic) root extraction and composition. The elementary functions are closed under [[derivative|differentiation]]. They are not closed under [[series (mathematics)|limits and infinite sums]]. Importantly, the elementary functions are {{em|not}} closed under [[antiderivative|integration]], as shown by [[Liouville's theorem (differential algebra)|Liouville's theorem]], see [[nonelementary integral]]. The [[Liouvillian function]]s are defined as the elementary functions and, recursively, the integrals of the Liouvillian functions.

==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'' is formalized in [[differential algebra]].  A [[differential field]] is a [[field (mathematics)|field]]  with an 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'' {{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. 

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.

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
* [[Algebraic function|algebraic]] over the preceding field, or
* 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 [[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.

==See also==

* [[Algebraic function]]
* {{anl|Closed-form expression}}
* [[Differential Galois theory]]
* {{anl|Elementary function arithmetic}}
* {{anl|Liouville's theorem (differential algebra)}}
* [[Tarski's high school algebra problem]]
* {{anl|Transcendental function}}

==Notes==
{{reflist}}

==References==
* {{cite book |last=Forsyth |first=Andrew |author-link=Andrew Forsyth |title=Theory of Functions of a Complex Variable |date=1893 |url=https://archive.org/details/theoryoffunction00fors/?q=region |publisher=Cambridge |jfm=25.0652.01 }}
*{{Cite journal
 | last = Liouville
 | first = Joseph
 | author-link = Joseph Liouville
 | title = Premier mémoire sur la détermination des intégrales dont la valeur est algébrique
 | journal = Journal de l'École Polytechnique
 | year = 1833a
 | volume = tome XIV
 | pages = 124–148
 | url = http://gallica.bnf.fr/ark:/12148/bpt6k433678n/f127.item.r=Liouville
}}
*{{Cite journal
 | last = Liouville
 | first = Joseph
 | author-link = Joseph Liouville
 | title = Second mémoire sur la détermination des intégrales dont la valeur est algébrique
 | journal = Journal de l'École Polytechnique
 | year = 1833b
 | volume = tome XIV
 | pages = 149–193
 | url = http://gallica.bnf.fr/ark:/12148/bpt6k433678n/f152.item.r=Liouville
}}
*{{Cite journal
 | last = Liouville
 | first = Joseph
 | author-link = Joseph Liouville
 | title = Note sur la détermination des intégrales dont la valeur est algébrique
 | journal = [[Journal für die reine und angewandte Mathematik]]
 | year = 1833c
 | volume = 10
 | pages = 347–359
 | url = http://gdz.sub.uni-goettingen.de/en/dms/loader/img/?PID=GDZPPN002139332
}}
*{{Cite book
 | last = Ritt
 | first = Joseph
 | author-link = Joseph Ritt
 | title = Differential Algebra
 | publisher = [[American Mathematical Society|AMS]]
 | year = 1950
 | url = https://www.ams.org/online_bks/coll33/
}}
*{{Cite journal
 | last = Rosenlicht
 | first = Maxwell
 | author-link = Maxwell Rosenlicht
 | title = Integration in finite terms
 | journal = [[American Mathematical Monthly]]
 | year = 1972
 | volume = 79
 | issue = 9
 | pages = 963–972
 | doi = 10.2307/2318066
 | jstor=2318066
}}

==Further reading==

* {{cite book |doi=10.1007/978-3-540-73086-6_5|chapter=What Might "Understand a Function" Mean? |title=Towards Mechanized Mathematical Assistants |series=Lecture Notes in Computer Science |year=2007 |last1=Davenport |first1=James H. |volume=4573 |pages=55–65 |isbn=978-3-540-73083-5|s2cid=8049737}}

==External links==
* [https://www.encyclopediaofmath.org/index.php/Elementary_functions ''Elementary functions'' at Encyclopaedia of Mathematics]
* {{MathWorld|ElementaryFunction|Elementary function}}

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[[Category:Differential algebra]]
[[Category:Computer algebra]]
[[Category:Types of functions]]