Editing Elementary function

{{short description|A kind of mathematical function}}
{{About| |the complexity class | Elementary recursive function}}

In [[mathematics]], an '''elementary function''' is a [[function (mathematics)|function]] of a single [[variable (mathematics)|variable]] (typically [[Function of a real variable|real]] or [[Complex analysis#Complex functions|complex]]) that is defined as taking [[addition|sums]], [[multiplication|products]], [[algebraic function|roots]] and [[composition of functions|compositions]] of [[finite set|finitely]] many [[Polynomial#Polynomial functions|polynomial]], [[Rational function|rational]], [[Trigonometric functions|trigonometric]], [[Hyperbolic functions|hyperbolic]], and [[Exponential function|exponential]] functions, and their [[Inverse function|inverses]] (e.g., [[Inverse trigonometric functions|arcsin]], [[Natural logarithm|log]], or ''x''<sup>1/''n''</sup>).<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>

All elementary functions are continuous on their [[Domain of a function|domains]].

Elementary functions were introduced by [[Joseph Liouville]] in a series of papers from 1833 to 1841.<ref>{{harvnb|Liouville|1833a}}.</ref><ref>{{harvnb|Liouville|1833b}}.</ref><ref>{{harvnb|Liouville|1833c}}.</ref>  An [[abstract algebra|algebraic]] treatment of elementary functions was started by [[Joseph Fels Ritt]] in the 1930s.<ref>{{harvnb|Ritt|1950}}.</ref> Many textbooks and dictionaries do not give a precise definition of the elementary functions, and mathematicians differ on it.<ref name=":0">{{Cite journal |last1=Subbotin |first1=Igor Ya. |last2=Bilotskii |first2=N. N. |date=March 2008 |title=Algorithms and Fundamental Concepts of Calculus |url=https://assets.nu.edu/assets/resources/pageResources/Journal_of_Research_March081.pdf |journal=Journal of Research in Innovative Teaching |volume=1 |issue=1 |pages=82–94}}</ref>

== Examples ==

=== Basic examples ===
Elementary functions of a single variable {{mvar|x}} include:
* [[Constant function]]s: <math>2,\ \pi,\ e,</math> etc.
* [[Exponentiation#Rational_exponents|Rational powers of {{mvar|x}}]]: <math>x,\ x^2,\ \sqrt{x}\ (x^\frac{1}{2}),\ x^\frac{2}{3},</math> etc.
* [[Exponential function]]s: <math>e^x, \ a^x</math>
* [[Logarithm]]s: <math>\log x, \ \log_a x</math>
* [[Trigonometric function]]s: <math>\sin x,\ \cos x,\ \tan 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|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 by root extraction of a polynomial with coefficients in elementary functions<ref name=":1" />
* 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. 

The [[Absolute value|absolute value function]], for real <math>x</math>, is also elementary<!-- {{cn|date=February 2021}} the proof is already provided here? --> as it can be expressed as the composition of a power and root of <math>x</math>: <math display="inline">|x|=\sqrt{x^2}</math>.{{dubious|date=June 2024}}

=== Non-elementary functions ===
Many mathematicians exclude non-[[Analytic function|analytic functions]] such as the [[Absolute value|absolute value function]] or discontinuous functions such as the [[step function]],<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><ref name=":0" /> but others allow them. Some have proposed extending the set 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>

Some examples of functions that are ''not'' elementary:

* [[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 ==
It follows directly from the definition that the set of elementary functions is [[closure (mathematics)|closed]] under arithmetic operations, 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==

The mathematical definition of an '''elementary function''', or a function in elementary form, is considered in the context of [[differential algebra]].  A differential algebra is an algebra with the 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''<sub>0</sub> (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>

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

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

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.

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==

* {{annotated link|Algebraic function}}
* {{annotated link|Closed-form expression}}
* {{annotated link|Differential Galois theory}}
* {{annotated link|Elementary function arithmetic}}
* {{annotated link|Liouville's theorem (differential algebra)}}
* {{annotated link|Tarski's high school algebra problem}}
* {{annotated link|Transcendental function}}
* {{annotated link|Tupper's self-referential formula}}

==Notes==
{{reflist}}

==References==
*{{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]]