Elementary function

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

• Constant functions: $2,\backslash \; \backslash pi,\backslash \; e,$ etc.
• Exponentiation#Rational_exponents: $x,\backslash \; x^2,\backslash \; \backslash sqrt\{x\}\backslash \; (x^\backslash frac\{1\}\{2\}),\backslash \; x^\backslash frac\{2\}\{3\},$ etc.
• Exponential functions: $e^x,\; \backslash \; a^x$
• Logarithms: $\backslash log\; x,\; \backslash \; \backslash log\_a\; x$
• Trigonometric functions: $\backslash sin\; x,\backslash \; \backslash cos\; x,\backslash \; \backslash tan\; x,$ etc.
• Inverse trigonometric functions: $\backslash arcsin\; x,\backslash \; \backslash arccos\; x,$ etc.
• Hyperbolic functions: $\backslash sinh\; x,\backslash \; \backslash cosh\; x,$ etc.
• Inverse hyperbolic functions: $\backslash operatorname\{arsinh\}\; x,\backslash \; \backslash operatorname\{arcosh\}\; x,$ etc.
• All functions obtained by adding, subtracting, multiplying or dividing a finite number of any of the previous functions{{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}}
• All functions obtained by root extraction of a polynomial with coefficients in elementary functions[https://mathworld.wolfram.com/ElementaryFunction.html Weisstein, Eric W. "Elementary Function." From MathWorld]
• All functions obtained by composing a finite number of any of the previously listed functions

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

Examples of elementary functions include:

• Addition, e.g. ({{mvar|x}} + 1)
• Multiplication, e.g. (2{{mvar|x}})
• Polynomial functions
• $\backslash frac\{e^\{\backslash tan\; x\}\}\{1+x^2\}\backslash sin\backslash left(\backslash sqrt\{1+(\backslash log\; x)^2\}\backslash right)$
• $-i\backslash log\backslash left(x+i\backslash sqrt\{1-x^2\}\backslash right)$

The last function is equal to $\backslash arccos\; x$, the inverse cosine, in the entire complex plane.

All monomials, polynomials, rational functions and algebraic functions are elementary.

The absolute value function, for real $x$, is also elementary as it can be expressed as the composition of a power and root of $x$: $|x|=\backslash sqrt\{x^2\}$.{{dubious|date=June 2024}}

Many mathematicians exclude non-analytic functions such as the absolute value function or discontinuous functions such as the step function,{{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}} but others allow them. Some have proposed extending the set to include, for example, the Lambert W function.{{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}}

Some examples of functions that are not elementary:

• tetration
• the gamma function
• non-elementary Liouvillian functions, including
• the exponential integral (Ei), logarithmic integral (Li or li) and Fresnel integrals (S and C).
• the error function, $\backslash mathrm\{erf\}(x)=\backslash frac\{2\}\{\backslash sqrt\{\backslash pi\}\}\backslash int\_0^x\; e^\{-t^2\}\backslash ,dt,$ a fact that may not be immediately obvious, but can be proven using the Risch algorithm.
• other nonelementary integrals, including the Dirichlet integral and elliptic integral.

It follows directly from the definition that the set of elementary functions is closed under arithmetic operations, root extraction and composition. The elementary functions are closed under differentiation. They are not closed under limits and infinite sums. Importantly, the elementary functions are {{em|not}} closed under integration, as shown by Liouville's theorem, see nonelementary integral. The Liouvillian functions are defined as the elementary functions and, recursively, the integrals of the Liouvillian functions.

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 extensions of the algebra. By starting with the field of rational functions, 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 rationals 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

: $\backslash partial\; (u\; +\; v)\; =\; \backslash partial\; u\; +\; \backslash partial\; v$

and satisfies the Leibniz product rule

: $\backslash partial(u\backslash cdot\; v)=\backslash partial\; u\backslash cdot\; v+u\backslash cdot\backslash partial\; v\backslash ,.$

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

• is algebraic over F, or
• is an exponential, that is, ∂u = u ∂a for a ∈ F, or
• is a logarithm, that is, ∂u = ∂a / a for a ∈ F.

(see also Liouville's theorem)

• {{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}}

{{reflist}}

• {{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 = 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

}}

• {{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}}

• [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