Elementary function

<h2 id="examples">Examples</h2>
<h3>Basic examples</h3>
Elementary functions of a single variable x include:

<ul><li><a href="/facts/Constant_function/BmPx6dn7">Constant functions</a>: 
 
 
 
 2
 ,
  
 π
 ,
  
 e
 ,
 
 
 {\displaystyle 2,\ \pi ,\ e,}
 
 etc.</li>
<li><a href="/facts/Exponentiation/pac6QY2g">Rational powers of x</a>: 
 
 
 
 x
 ,
  
 
 x
 
 2
 
 
 ,
  
 
 
 x
 
 
  
 (
 
 x
 
 
 1
 2
 
 
 
 )
 ,
  
 
 x
 
 
 2
 3
 
 
 
 ,
 
 
 {\displaystyle x,\ x^{2},\ {\sqrt {x}}\ (x^{\frac {1}{2}}),\ x^{\frac {2}{3}},}
 
 etc.</li>
<li><a href="/facts/Exponential_function/ewlYxvR2">Exponential functions</a>: 
 
 
 
 
 e
 
 x
 
 
 ,
  
 
 a
 
 x
 
 
 
 
 {\displaystyle e^{x},\ a^{x}}
 
</li>
<li><a href="/facts/Logarithm/QeXaV97q">Logarithms</a>: 
 
 
 
 log
 ⁡
 x
 ,
  
 
 log
 
 a
 
 
 ⁡
 x
 
 
 {\displaystyle \log x,\ \log _{a}x}
 
</li>
<li><a href="/facts/Trigonometric_function/czej7ur0">Trigonometric functions</a>: 
 
 
 
 sin
 ⁡
 x
 ,
  
 cos
 ⁡
 x
 ,
  
 tan
 ⁡
 x
 ,
 
 
 {\displaystyle \sin x,\ \cos x,\ \tan x,}
 
 etc.</li>
<li><a href="/facts/Inverse_trigonometric_function/bVwU9FPG">Inverse trigonometric functions</a>: 
 
 
 
 arcsin
 ⁡
 x
 ,
  
 arccos
 ⁡
 x
 ,
 
 
 {\displaystyle \arcsin x,\ \arccos x,}
 
 etc.</li>
<li><a href="/facts/Hyperbolic_function/Y4Cb5S35">Hyperbolic functions</a>: 
 
 
 
 sinh
 ⁡
 x
 ,
  
 cosh
 ⁡
 x
 ,
 
 
 {\displaystyle \sinh x,\ \cosh x,}
 
 etc.</li>
<li><a href="/facts/Inverse_hyperbolic_function/myk0wJty">Inverse hyperbolic functions</a>: 
 
 
 
 arsinh
 ⁡
 x
 ,
  
 arcosh
 ⁡
 x
 ,
 
 
 {\displaystyle \operatorname {arsinh} x,\ \operatorname {arcosh} x,}
 
 etc.</li>
<li>All functions obtained by adding, subtracting, multiplying or dividing a finite number of any of the previous functions<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a></li>
<li>All functions obtained by root extraction of a polynomial with coefficients in elementary functions<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a></li>
<li>All functions obtained by <a href="/facts/Function_composition/r5Pvnmth">composing</a> a finite number of any of the previously listed functions</li></ul>
Certain elementary functions of a single complex variable z, such as 
 
 
 
 
 
 z
 
 
 
 
 {\displaystyle {\sqrt {z}}}
 
 and 
 
 
 
 log
 ⁡
 z
 
 
 {\displaystyle \log z}
 
, may be <a href="/facts/Multivalued_function/WH5ua6BL">multivalued</a>. Additionally, certain classes of functions may be obtained by others using the final two rules. For example, the exponential function 
 
 
 
 
 e
 
 z
 
 
 
 
 {\displaystyle e^{z}}
 
 composed with addition, subtraction, and division provides the hyperbolic functions, while initial composition with 
 
 
 
 i
 z
 
 
 {\displaystyle iz}
 
 instead provides the trigonometric functions.

<h3>Composite examples</h3>
Examples of elementary functions include:

<ul><li>Addition, e.g. (x + 1)</li>
<li>Multiplication, e.g. (2x)</li>
<li><a href="/facts/Polynomial/Lzak8VVx">Polynomial</a> functions</li>
<li>
 
 
 
 
 
 
 e
 
 tan
 ⁡
 x
 
 
 
 1
 +
 
 x
 
 2
 
 
 
 
 
 sin
 ⁡
 
 (
 
 
 1
 +
 (
 log
 ⁡
 x
 
 )
 
 2
 
 
 
 
 )
 
 
 
 {\displaystyle {\frac {e^{\tan x}}{1+x^{2}}}\sin \left({\sqrt {1+(\log x)^{2}}}\right)}
 
</li>
<li>
 
 
 
 −
 i
 log
 ⁡
 
 (
 
 x
 +
 i
 
 
 1
 −
 
 x
 
 2
 
 
 
 
 
 )
 
 
 
 {\displaystyle -i\log \left(x+i{\sqrt {1-x^{2}}}\right)}
 
</li></ul>
The last function is equal to 
 
 
 
 arccos
 ⁡
 x
 
 
 {\displaystyle \arccos x}
 
, the <a href="/facts/Inverse_trigonometric_functions/bVwU9FPG">inverse cosine</a>, in the entire <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a>.
All <a href="/facts/Monomial/6VAPCYXJ">monomials</a>, <a href="/facts/Polynomial/Lzak8VVx">polynomials</a>, <a href="/facts/Rational_function/2nJGu0Ko">rational functions</a> and <a href="/facts/Algebraic_function/yT7EKmwF">algebraic functions</a> are elementary. 
The <a href="/facts/Absolute_value/dwJBpEs5">absolute value function</a>, for real 
 
 
 
 x
 
 
 {\displaystyle x}
 
, is also elementary as it can be expressed as the composition of a power and root of 
 
 
 
 x
 
 
 {\displaystyle x}
 
: 
 
 
 
 
 |
 
 x
 
 |
 
 =
 
 
 
 x
 
 2
 
 
 
 
 
 
 {\textstyle |x|={\sqrt {x^{2}}}}
 
.[dubious – discuss]

<h3>Non-elementary functions</h3>
Many mathematicians exclude non-<a href="/facts/Analytic_function/JhL3z8FP">analytic functions</a> such as the <a href="/facts/Absolute_value/dwJBpEs5">absolute value function</a> or discontinuous functions such as the <a href="/facts/Step_function/VnrEjo4B">step function</a>,<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a><a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> but others allow them. Some have proposed extending the set to include, for example, the <a href="/facts/Lambert_W_function/NG1wd71f">Lambert W function</a>.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>
Some examples of functions that are not elementary:

<ul><li><a href="/facts/Tetration/5rKeVVYv">tetration</a></li>
<li>the <a href="/facts/Gamma_function/SjGQcnyw">gamma function</a></li>
<li>non-elementary <a href="/facts/Liouvillian_function/s5dL3bR8">Liouvillian functions</a>, including
<ul><li>the <a href="/facts/Exponential_integral/JF6aMcmL">exponential integral</a> (Ei), <a href="/facts/Logarithmic_integral/P3xwqRlA">logarithmic integral</a> (Li or li) and <a href="/facts/Fresnel_integral/5W7z760j">Fresnel integrals</a> (S and C).</li>
<li>the <a href="/facts/Error_function/Ca2H7pGr">error function</a>, 
 
 
 
 
 e
 r
 f
 
 (
 x
 )
 =
 
 
 2
 
 π
 
 
 
 
 ∫
 
 0
 
 
 x
 
 
 
 e
 
 −
 
 t
 
 2
 
 
 
 
 
 d
 t
 ,
 
 
 {\displaystyle \mathrm {erf} (x)={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}\,dt,}
 
 a fact that may not be immediately obvious, but can be proven using the <a href="/facts/Risch_algorithm/UcT2qrPl">Risch algorithm</a>.</li></ul></li>
<li>other <a href="/facts/Nonelementary_integral/IVp2vzq7">nonelementary integrals</a>, including the <a href="/facts/Dirichlet_integral/fFo9lHcb">Dirichlet integral</a> and <a href="/facts/Elliptic_integral/q8EGYzQ3">elliptic integral</a>.</li></ul>
<h2 id="closure">Closure</h2>
It follows directly from the definition that the set of elementary functions is <a href="/facts/Closure_(mathematics)/Ct6r2fJz">closed</a> under arithmetic operations, root extraction and composition. The elementary functions are closed under <a href="/facts/Derivative/7Ito70Dh">differentiation</a>. They are not closed under <a href="/facts/Series_(mathematics)/yDnlsgIe">limits and infinite sums</a>. Importantly, the elementary functions are not closed under <a href="/facts/Antiderivative/5qDsudIv">integration</a>, as shown by <a href="/facts/Liouville%27s_theorem_(differential_algebra)/w9tx70G7">Liouville's theorem</a>, see <a href="/facts/Nonelementary_integral/IVp2vzq7">nonelementary integral</a>. The <a href="/facts/Liouvillian_function/s5dL3bR8">Liouvillian functions</a> are defined as the elementary functions and, recursively, the integrals of the Liouvillian functions.

<h2 id="differential-algebra">Differential algebra</h2>
The mathematical definition of an elementary function, or a function in elementary form, is considered in the context of <a href="/facts/Differential_algebra/CXbpIMgk">differential algebra</a>. 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 <a href="/facts/Field_extension/L7yIDtyK">extensions</a> of the algebra. By starting with the <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> of <a href="/facts/Rational_function/2nJGu0Ko">rational functions</a>, 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 F0 (rational functions over the <a href="/facts/Rational_number/VJQmyY05">rationals</a> Q for example) together with a derivation map u → ∂u. (Here ∂u is a new function. Sometimes the notation u′ is used.) The derivation captures the properties of differentiation, so that for any two elements of the base field, the derivation is linear

∂
        (
        u
        +
        v
        )
        =
        ∂
        u
        +
        ∂
        v
      
    
    {\displaystyle \partial (u+v)=\partial u+\partial v}

and satisfies the <a href="/facts/Product_rule/MyKxsP5t">Leibniz product rule</a>

∂
        (
        u
        ⋅
        v
        )
        =
        ∂
        u
        ⋅
        v
        +
        u
        ⋅
        ∂
        v
        
        .
      
    
    {\displaystyle \partial (u\cdot v)=\partial u\cdot v+u\cdot \partial v\,.}

An element h is a constant if ∂h = 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

<ul><li>is <a href="/facts/Algebraic_function/yT7EKmwF">algebraic</a> over F, or</li>
<li>is an exponential, that is, ∂u = u ∂a for a ∈ F, or</li>
<li>is a logarithm, that is, ∂u = ∂a / a for a ∈ F.</li></ul>
(see also <a href="/facts/Liouville%27s_theorem_(differential_algebra)/w9tx70G7">Liouville's theorem</a>)

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Algebraic_function/yT7EKmwF">Algebraic function</a> – Mathematical function</li>
<li><a href="/facts/Closed-form_expression/y0xCXlSk">Closed-form expression</a> – Mathematical formula involving a given set of operations</li>
<li><a href="/facts/Differential_Galois_theory/neKl6CLC">Differential Galois theory</a> – Study of Galois symmetry groups of differential fields</li>
<li><a href="/facts/Elementary_function_arithmetic/mDBRMjpw">Elementary function arithmetic</a> – System of arithmetic in proof theory</li>
<li><a href="/facts/Liouville%27s_theorem_(differential_algebra)/w9tx70G7">Liouville's theorem (differential algebra)</a> – Says when antiderivatives of elementary functions can be expressed as elementary functions</li>
<li><a href="/facts/Tarski%27s_high_school_algebra_problem/DmvEPAQx">Tarski's high school algebra problem</a> – Mathematical problem</li>
<li><a href="/facts/Transcendental_function/yMkeq4Eo">Transcendental function</a> – Analytic function that does not satisfy a polynomial equation</li>
<li><a href="/facts/Tupper%27s_self-referential_formula/3R7nkEh1">Tupper's self-referential formula</a> – Formula that visually represents itself when graphed</li></ul>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Joseph_Liouville/U22o7wtK">Liouville, Joseph</a> (1833a). <a href="http://gallica.bnf.fr/ark:/12148/bpt6k433678n/f127.item.r=Liouville">"Premier mémoire sur la détermination des intégrales dont la valeur est algébrique"</a>. Journal de l'École Polytechnique. tome XIV: 124–148.</li>
<li><a href="/facts/Joseph_Liouville/U22o7wtK">Liouville, Joseph</a> (1833b). <a href="http://gallica.bnf.fr/ark:/12148/bpt6k433678n/f152.item.r=Liouville">"Second mémoire sur la détermination des intégrales dont la valeur est algébrique"</a>. Journal de l'École Polytechnique. tome XIV: 149–193.</li>
<li><a href="/facts/Joseph_Liouville/U22o7wtK">Liouville, Joseph</a> (1833c). <a href="http://gdz.sub.uni-goettingen.de/en/dms/loader/img/?PID=GDZPPN002139332">"Note sur la détermination des intégrales dont la valeur est algébrique"</a>. <a href="/facts/Journal_f%C3%BCr_die_reine_und_angewandte_Mathematik/YCcRSaAQ">Journal für die reine und angewandte Mathematik</a>. 10: 347–359.</li>
<li><a href="/facts/Joseph_Ritt/4ChIswSR">Ritt, Joseph</a> (1950). <a href="https://www.ams.org/online_bks/coll33/">Differential Algebra</a>. <a href="/facts/American_Mathematical_Society/fr5gotCt">AMS</a>.</li>
<li><a href="/facts/Maxwell_Rosenlicht/1FaHNBYS">Rosenlicht, Maxwell</a> (1972). "Integration in finite terms". <a href="/facts/American_Mathematical_Monthly/Te6CHvkg">American Mathematical Monthly</a>. 79 (9): 963–972. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2318066">10.2307/2318066</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2318066">2318066</a>.</li></ul>
<h2 id="further-reading">Further reading</h2>
<ul><li>Davenport, James H. (2007). "What Might "Understand a Function" Mean?". Towards Mechanized Mathematical Assistants. Lecture Notes in Computer Science. Vol. 4573. pp. 55–65. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-540-73086-6_5">10.1007/978-3-540-73086-6_5</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-73083-5. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:8049737">8049737</a>.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="https://www.encyclopediaofmath.org/index.php/Elementary_functions">Elementary functions at Encyclopaedia of Mathematics</a></li>
<li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/ElementaryFunction.html">"Elementary function"</a>. <a href="/facts/MathWorld/yc1jodbq">MathWorld</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Spivak, Michael. (1994). Calculus (3rd ed.). Houston, Tex.: Publish or Perish. p. 359. ISBN 0914098896. OCLC 31441929. <a href="0914098896" target="_blank">0914098896</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Liouville 1833a. - Liouville, Joseph (1833a). "Premier mémoire sur la détermination des intégrales dont la valeur est algébrique". Journal de l'École Polytechnique. tome XIV: 124–148. <a href="http://gallica.bnf.fr/ark:/12148/bpt6k433678n/f127.item.r=Liouville" target="_blank">http://gallica.bnf.fr/ark:/12148/bpt6k433678n/f127.item.r=Liouville</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Liouville 1833b. - Liouville, Joseph (1833b). "Second mémoire sur la détermination des intégrales dont la valeur est algébrique". Journal de l'École Polytechnique. tome XIV: 149–193. <a href="http://gallica.bnf.fr/ark:/12148/bpt6k433678n/f152.item.r=Liouville" target="_blank">http://gallica.bnf.fr/ark:/12148/bpt6k433678n/f152.item.r=Liouville</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Liouville 1833c. - Liouville, Joseph (1833c). "Note sur la détermination des intégrales dont la valeur est algébrique". Journal für die reine und angewandte Mathematik. 10: 347–359. <a href="http://gdz.sub.uni-goettingen.de/en/dms/loader/img/?PID=GDZPPN002139332" target="_blank">http://gdz.sub.uni-goettingen.de/en/dms/loader/img/?PID=GDZPPN002139332</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Ritt 1950. - Ritt, Joseph (1950). Differential Algebra. AMS. <a href="https://www.ams.org/online_bks/coll33/" target="_blank">https://www.ams.org/online_bks/coll33/</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Subbotin, Igor Ya.; Bilotskii, N. N. (March 2008). "Algorithms and Fundamental Concepts of Calculus" (PDF). Journal of Research in Innovative Teaching. 1 (1): 82–94. <a href="https://assets.nu.edu/assets/resources/pageResources/Journal_of_Research_March081.pdf" target="_blank">https://assets.nu.edu/assets/resources/pageResources/Journal_of_Research_March081.pdf</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Ordinary Differential Equations. Dover. 1985. p. 17. ISBN 0-486-64940-7. <a href="0-486-64940-7" target="_blank">0-486-64940-7</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Weisstein, Eric W. "Elementary Function." From MathWorld <a href="https://mathworld.wolfram.com/ElementaryFunction.html" target="_blank">https://mathworld.wolfram.com/ElementaryFunction.html</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Risch, Robert H. (1979). "Algebraic Properties of the Elementary Functions of Analysis". American Journal of Mathematics. 101 (4): 743–759. doi:10.2307/2373917. ISSN 0002-9327. JSTOR 2373917. <a href="https://www.jstor.org/stable/2373917" target="_blank">https://www.jstor.org/stable/2373917</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Subbotin, Igor Ya.; Bilotskii, N. N. (March 2008). "Algorithms and Fundamental Concepts of Calculus" (PDF). Journal of Research in Innovative Teaching. 1 (1): 82–94. <a href="https://assets.nu.edu/assets/resources/pageResources/Journal_of_Research_March081.pdf" target="_blank">https://assets.nu.edu/assets/resources/pageResources/Journal_of_Research_March081.pdf</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Stewart, Seán (2005). "A new elementary function for our curricula?" (PDF). Australian Senior Mathematics Journal. 19 (2): 8–26. <a href="https://files.eric.ed.gov/fulltext/EJ720055.pdf" target="_blank">https://files.eric.ed.gov/fulltext/EJ720055.pdf</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
</ol>

Elementary function open-in-new

Elementary function