Elementary function: Difference between revisions
simplify lead; clarify abs; note that definitions vary Tags: Reverted Visual edit |
Undoing edits concerning absolute value. See Talk page. |
||
Line 2: | Line 2: | ||
{{About| |the complexity class |ELEMENTARY| the logical system | Elementary function arithmetic}} |
{{About| |the complexity class |ELEMENTARY| the logical system | Elementary function arithmetic}} |
||
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, |
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, including possibly their [[Inverse function|inverse functions]] (e.g., [[Inverse trigonometric functions|arcsin]], [[Natural logarithm|log]], or ''x''<sup>1/''n''</sup>).<ref>{{Cite book|title=Calculus|last=Spivak, Michael.|date=1994|publisher=Publish or Perish|isbn=0914098896|edition=3rd|location=Houston, Tex.|pages=359|oclc=31441929}}</ref> |
||
All elementary functions are continuous on their [[Domain of a function|domains]]. |
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}}. |
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> |
||
== Examples == |
== Examples == |
||
Line 37: | Line 37: | ||
The last function is equal to <math>\arccos x</math>, the [[Inverse_trigonometric_functions#Logarithmic_forms|inverse cosine]], in the entire [[complex plane]]. |
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. |
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>. |
||
=== Non-elementary functions === |
=== Non-elementary functions === |
||
Most 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 |issn=0002-9327}}</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: |
Some examples of functions that are ''not'' elementary: |
||
Line 48: | Line 46: | ||
* non-elementary [[Liouvillian function#Examples|Liouvillian functions]], including |
* non-elementary [[Liouvillian function#Examples|Liouvillian functions]], including |
||
** the [[Exponential integral|exponential]] (''Ei''), [[logarithmic integral]] (''Li'' or ''li'') and [[Fresnel integral|Fresnel]] (''S'' and ''C'') integrals. |
** the [[Exponential integral|exponential]] (''Ei''), [[logarithmic integral]] (''Li'' or ''li'') and [[Fresnel integral|Fresnel]] (''S'' and ''C'') integrals. |
||
** 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, |
** 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 [[ |
* other [[Nonelementary integral]]s, including the [[Dirichlet integral]] and [[elliptic integral]]. |
||
== Closure == |
== 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 [[ |
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== |
==Differential algebra== |
Revision as of 13:14, 9 June 2024
In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponential functions, including possibly their inverse functions (e.g., arcsin, log, or x1/n).[1]
All elementary functions are continuous on their domains.
Elementary functions were introduced by Joseph Liouville in a series of papers from 1833 to 1841.[2][3][4] An algebraic treatment of elementary functions was started by Joseph Fels Ritt in the 1930s.[5]
Examples
Basic examples
Elementary functions of a single variable x include:
- Constant functions: etc.
- Rational powers of x: etc.
- Exponential functions:
- Logarithms:
- Trigonometric functions: etc.
- Inverse trigonometric functions: etc.
- Hyperbolic functions: etc.
- Inverse hyperbolic functions: etc.
- All functions obtained by adding, subtracting, multiplying or dividing a finite number of any of the previous functions[6]
- All functions obtained by root extraction of a polynomial with coefficients in elementary functions[7]
- All functions obtained by composing a finite number of any of the previously listed functions
Certain elementary functions of a single complex variable z, such as and , may be multivalued. Additionally, certain classes of functions may be obtained by others using the final two rules. For example, the exponential function composed with addition, subtraction, and division provides the hyperbolic functions, while initial composition with instead provides the trigonometric functions.
Composite examples
Examples of elementary functions include:
- Addition, e.g. (x+1)
- Multiplication, e.g. (2x)
- Polynomial functions
The last function is equal to , the inverse cosine, in the entire complex plane.
All monomials, polynomials, rational functions and algebraic functions are elementary. The absolute value function, for real , is also elementary as it can be expressed as the composition of a power and root of : .
Non-elementary functions
Some examples of functions that are not elementary:
- tetration
- the gamma function
- non-elementary Liouvillian functions, including
- the exponential (Ei), logarithmic integral (Li or li) and Fresnel (S and C) integrals.
- the error function, 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.
Closure
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 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.
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 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
and satisfies the Leibniz product rule
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)
See also
- Algebraic function – Mathematical function
- Closed-form expression – Mathematical formula involving a given set of operations
- Differential Galois theory – Study of Galois symmetry groups of differential fields
- Elementary function arithmetic – System of arithmetic in proof theory
- Liouville's theorem (differential algebra) – Says when antiderivatives of elementary functions can be expressed as elementary functions
- Tarski's high school algebra problem – Mathematical problem
- Transcendental function – Analytic function that does not satisfy a polynomial equation
- Tupper's self-referential formula – Formula that visually represents itself when graphed
Notes
- ^ Spivak, Michael. (1994). Calculus (3rd ed.). Houston, Tex.: Publish or Perish. p. 359. ISBN 0914098896. OCLC 31441929.
- ^ Liouville 1833a.
- ^ Liouville 1833b.
- ^ Liouville 1833c.
- ^ Ritt 1950.
- ^ Ordinary Differential Equations. Dover. 1985. p. 17. ISBN 0-486-64940-7.
- ^ Weisstein, Eric W. "Elementary Function." From MathWorld
References
- Liouville, Joseph (1833a). "Premier mémoire sur la détermination des intégrales dont la valeur est algébrique". Journal de l'École Polytechnique. tome XIV: 124–148.
- Liouville, Joseph (1833b). "Second mémoire sur la détermination des intégrales dont la valeur est algébrique". Journal de l'École Polytechnique. tome XIV: 149–193.
- Liouville, Joseph (1833c). "Note sur la détermination des intégrales dont la valeur est algébrique". Journal für die reine und angewandte Mathematik. 10: 347–359.
- Ritt, Joseph (1950). Differential Algebra. AMS.
- Rosenlicht, Maxwell (1972). "Integration in finite terms". American Mathematical Monthly. 79 (9): 963–972. doi:10.2307/2318066. JSTOR 2318066.
Further reading
- Davenport, James H. (2007). "What Might "Understand a Function" Mean?". Towards Mechanized Mathematical Assistants. Lecture Notes in Computer Science. Vol. 4573. pp. 55–65. doi:10.1007/978-3-540-73086-6_5. ISBN 978-3-540-73083-5. S2CID 8049737.