Editing Elementary function (section)

== 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]].