Elementary function: Difference between revisions

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Template:Short description Script error: No such module "Distinguish". In mathematics, an elementary function is a function of a single variable (real or complex) that is typically encountered by beginners. The basic elementary functions are polynomial functions, rational functions, the trigonometric functions, the exponential and logarithm functions, the n-th root, and the inverse trigonometric functions, as well as those functions obtained by addition, multiplication, division, and composition of these. Some functions which are encountered by beginners are not elementary, such as piecewise-defined functions. More generally, in some modern treatments, elementary functions comprise the set of functions previously enumerated, all algebraic functions, and all functions obtained by roots of a polynomial whose coefficients are elementary.

The elementary functions were originally defined by Joseph Liouville in 1833. A key property is that all elementary functions have derivatives of any order, which are also elementary, and can be algorithmically computed by applying the differentiation rules (or the rules for implicit differentiation in the case of roots). The Taylor series of an elementary function converges in a neighborhood of every point of its domain. More generally, they are global analytic functions, defined (possibly with multiple values, such as the elementary function $\sqrt{z}$ or $\log z$ ) for every complex argument, except at isolated points. In contrast, antiderivatives of elementary functions need not be elementary and is difficult to decide whether a specific elementary function has an elementary antiderivative.

Liouville's result is that, if an elementary function has an elementary antiderivative, then this antiderivative is a linear combination of logarithms, where the coefficients and the arguments of the logarithms are elementary functions involved, in some sense, in the definition of the function. The Risch algorithm (1968) can decide whether an elementary function has an elementary antiderivative, and, if so, to compute it. However, as of 2025^[update]Script error: No such module "Check for unknown parameters"., there is no full implementation.^[1]

Examples

Basic examples

Elementary functions of a single variable Template:Tmath include:

Constant functions: Template:Tmath, Template:Tmath, Template:Tmath, the Euler–Mascheroni constant, Apéry's constant, Khinchin's constant, etc. Any constant real (or complex) number.
[[Exponentiation|Powers of Template:Tmath]]: Template:Tmath, etc. (The exponent can be any real or complex constant.)
Exponential functions: Template:Tmath, Template:Tmath
Logarithms: Template:Tmath, Template:Tmath
Trigonometric functions: Template:Tmath, Template:Tmath, Template:Tmath, etc.
Inverse trigonometric functions: Template:Tmath, Template:Tmath, etc.
Hyperbolic functions: Template:Tmath, Template:Tmath, etc.
Inverse hyperbolic functions: Template:Tmath, Template:Tmath, etc.
All functions obtained by adding, subtracting, multiplying or dividing a finite number of any of the previous functions^[2]
All functions obtained as roots of a polynomial whose coefficients are elementary functions^[3]^[4]
All functions obtained by composing a finite number of any of the previously listed functions

Certain elementary functions of a single complex variable Template:Tmath, such as $\sqrt{z}$ and Template:Tmath, 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 $i z$ instead provides the trigonometric functions.

Composite examples

Examples of elementary functions include:

Addition, e.g. (Template:Tmath)
Multiplication, e.g. (Template:Tmath)
Polynomial functions
$- i \log (x + i \sqrt{1 - x^{2}})$

The last function is equal to Template:Tmath, the inverse cosine, in the entire complex plane.

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

Non-elementary functions

All elementary functions are analytic in the following sense: they can be extended to functions of a complex variable (possibly multivalued) that are analytic except at isolated points of the complex plane.^[5] Thus nonanalytic functions such as the absolute value function are not elementary,^[6] nor are most other piecewise-defined functions.

Not every analytic function is elementary. In fact, most special functions are not elementary. Non-elementary functions include:

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, Template:Tmath, 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.

Real-variables and analytic branches

In elementary real-variable settings such as those in calculus and pre-calculus, expressions involving roots, logarithms, and inverse trigonometric functions are often interpreted using fixed real branches on specified real domains. This convention is distinct from the analytic convention used in the theory of elementary functions and integration in finite terms. Risch gives a precise definition of elementary functions "in the sense of analysis" by using functions of a complex variable rather than a real variable. In this setting the elementary functions are built using algebraic operations, exponentials, and logarithms, and are represented in differential fields of meromorphic functions on regions of the complex plane or on Riemann surfaces.Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".

An algebraic equation such as $y^{2} = x^{2}$ has the local analytic branches $y = x$ and $y = - x$ . The real identity $\sqrt{x^{2}} = | x |$ uses the convention that $\sqrt{\cdot}$ denotes the nonnegative real square root, and so changes from one analytic branch to the other at $x = 0$ . Thus the restrictions of $| x |$ to $(0, \infty)$ and $(- \infty, 0)$ are elementary, but the usual real absolute value function on an interval containing $0$ is not a single analytic branch.

The same distinction appears in complex analysis. Whittaker and Watson note that although $x - i y$ and $| z |$ , where $z = x + i y$ , are functions of $z$ in a general sense, they are not elementary functions of the analytic type under consideration.Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters". In symbolic computation, functions such as absolute value, signum, and piecewise-defined functions can be treated instead by adjoining a step or conditional operation, which forms a separate class of piecewise function rings.Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".

Closure

It follows directly from the definition that the set of elementary functions is closed under arithmetic operations, (algebraic) 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 Template:Em 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.

Extensions

In late-nineteenth-century analysis, elementary functions were often classified into successive kinds according to the number of independent integrations required for their definition. Functions expressible without any integration—those generated from rational functions by algebraic operations together with exponentiation, logarithms, and circular or hyperbolic trigonometric functions—were said to be elementary functions of the first kind (in the sense of Liouville). Functions defined by a single integration of an algebraic function, such as the error function and the elliptic integrals, were elementary functions of the second kind; their inverses, the elliptic functions, were considered of the same order. Higher "kinds" (third, fourth, etc.) corresponded to multiple integrals of algebraic functions, giving rise to hyperelliptic and more general Abelian functions.Script error: No such module "Footnotes".Script error: No such module "Check for unknown parameters".

The essential point of the classification was that the class of elementary functions of any given kind be closed under the elementary operations—addition, multiplication, composition, and differentiation—so that differentiation never leads outside the same class, while integration may ascend to the next higher kind.

More recently, some have proposed extending the set of elementary functions by extending with certain transcendental functions, to include, for example, the Lambert W function^[7] or elliptic functions,^[8] all of which are analytic. The key attribute, from the perspective of the Liouville theorem, is that as a class, they are closed under taking derivatives. For example, the Lambert function $w = W (z)$ , which is defined implicitly by the equation Template:Tmath, has a derivative which can be obtained by implicit differentiation:

W^{'} (z) = \frac{e^{- W (z)}}{1 + W (z)},

which is again "elementary", provided that $W (z)$ is.

Differential algebra

The mathematical definition of an elementary function is formalized in differential algebra. A differential field is a field with an 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 Template:Tmath is a field together with a derivation Template:Tmath that maps Template:Tmath to itself. The derivation generalizes derivative, being linear (that is, Template:Tmath) and satisfying the Leibniz product rule (that is,Template:Tmath) for every two elements Template:Tmath and Template:Tmath in Template:Tmath. The rational functions over Template:Tmath of Template:Tmath form a basic examples of differential fields, when equipped with the usual derivative.

An element Template:Tmath of Template:Tmath is a constant if Template:Tmath. The constants of Template:Tmath form a differential field with zero derivative. Care must be taken that a differential field extension of a differential field may enlarge the field of constants.

A function Template:Tmath of a differential extension Template:Tmath of a differential field Template:Tmath is an elementary function over Template:Tmath if it belongs to a finite chain (for inclusion) of differential subfields of Template:Tmath that starts from Template:Tmath and is such that each is generated over the preceding one by a function that is either

algebraic over the preceding field, or
an exponential, that is, Template:Tmath for some Template:Mvar belonging to a prior subfield, or
a logarithm, that is, Template:Tmath for some Template:Mvar belonging to a prior subfield, (see Liouville's theorem)

With this definition, the usual elementary functions are exactly the function that are elementary over the field of the rational functions. This generalized definition allows considering every transcendental function as elementary for applying Liouville's theorem.

Notes

↑ Script error: No such module "citation/CS1".
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↑ Ritt, chapter 1
↑ Script error: No such module "Citation/CS1".
↑ Watson and Whittaker 1927, footnote to p 82. In the context of elementary functions, the function $y = f (x)$ defined as the root of $y^{2} - x^{2} = 0$ is two-valued: $y = \pm x$ .
↑ Script error: No such module "Citation/CS1".
↑ Ince, E. L. (1956) [1926]. Ordinary Differential Equations. New York: Dover Publications. ISBN 0-486-60339-4, footnote to p 330

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References

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External links

Elementary functions at Encyclopaedia of Mathematics
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[1] Script error: No such module "citation/CS1".

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[4] Ritt, chapter 1

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[6] Watson and Whittaker 1927, footnote to p 82. In the context of elementary functions, the function $y = f (x)$ defined as the root of $y^{2} - x^{2} = 0$ is two-valued: $y = \pm x$ .

[7] Script error: No such module "Citation/CS1".

[8] Ince, E. L. (1956) [1926]. Ordinary Differential Equations. New York: Dover Publications. ISBN 0-486-60339-4, footnote to p 330

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@@ Line 1: / Line 1: @@
-{{Short description|Type of mathematical function}}
+{{short description|Type of mathematical function}}
 {{distinguish|Elementary recursive function}}
-In [[mathematics]], an '''elementary function''' is a [[function (mathematics)|function]] of a single [[variable (mathematics)|variable]] ([[Function of a real variable|real]] or [[Complex analysis#Complex functions|complex]]) that is typically encountered by beginners.  The basic elementary functions are [[polynomial function]]s, [[rational function]]s, the [[trigonometric function]]s, the [[exponential function|exponential]] and [[logarithm]] functions, the [[n-th root]], and the [[inverse trigonometric function]]s, as well as those functions obtained by [[addition]], [[multiplication]], [[division (mathematics)|division]], and [[function composition|composition]] of these.  Some functions which are encountered by beginners are ''not'' elementary, such as the [[absolute value]] function and [[piecewise-defined function]]s. More generally, in modern mathematics, elementary functions comprise the set of functions previously enumerated, all [[algebraic function]]s (not often encountered by beginners), and all functions obtained by [[roots of a polynomial]] whose coefficients are elementary.
+In [[mathematics]], an '''elementary function''' is a [[function (mathematics)|function]] of a single [[variable (mathematics)|variable]] ([[Function of a real variable|real]] or [[Complex analysis#Complex functions|complex]]) that is typically encountered by beginners.  The basic elementary functions are [[polynomial function]]s, [[rational function]]s, the [[trigonometric function]]s, the [[exponential function|exponential]] and [[logarithm]] functions, the [[n-th root]], and the [[inverse trigonometric function]]s, as well as those functions obtained by [[addition]], [[multiplication]], [[division (mathematics)|division]], and [[function composition|composition]] of these.  Some functions which are encountered by beginners are ''not'' elementary, such as [[piecewise-defined function]]s. More generally, in some modern treatments, elementary functions comprise the set of functions previously enumerated, all [[algebraic function]]s, and all functions obtained by [[roots of a polynomial]] whose coefficients are elementary.
-This list of elementary functions was originally set forth by [[Joseph Liouville]] in 1833. A key property is that all elementary functions have [[derivative]]s of any order, which are also elementary, and can be [[algorithmically]] computed by applying the [[differentiation rules]] (or the rules for [[implicit differentiation]] in the case of roots). The [[Taylor series]] of an elementary function converges in a neighborhood of every point of its domain.  More generally, they are [[global analytic function]]s, defined (possibly with [[multivalued function|multiple values]], such as the elementary function <math>\sqrt z</math> or <math>\log z</math>) for every [[complex number|complex]] argument, except at [[isolated point]]s. In contrast, [[antiderivative]]s of elementary functions need not be elementary and is difficult to decide whether a specific elementary function has an elementary antiderivative.
+The elementary functions were originally defined by [[Joseph Liouville]] in 1833. A key property is that all elementary functions have [[derivative]]s of any order, which are also elementary, and can be [[algorithmically]] computed by applying the [[differentiation rules]] (or the rules for [[implicit differentiation]] in the case of roots). The [[Taylor series]] of an elementary function converges in a neighborhood of every point of its domain.  More generally, they are [[global analytic function]]s, defined (possibly with [[multivalued function|multiple values]], such as the elementary function <math>\sqrt z</math> or <math>\log z</math>) for every [[complex number|complex]] argument, except at [[isolated point]]s. In contrast, [[antiderivative]]s of elementary functions need not be elementary and is difficult to decide whether a specific elementary function has an elementary antiderivative.
-[[Liouville's theorem (differential algebra)|Liouville's result]] is that, if an elementary function has an elementary antiderivative, then this antiderivative is a linear combination of logarithms, where the coefficients and the arguments of the logarithms are elementary functions involved, in some sense, in the definition of the function. More than 130 years later, [[Risch algorithm]], named after [[Robert Henry Risch]], is an algorithm to decide whether an elementary function has an elementary antiderivative, and, if it has, to compute this antiderivative. Despite dealing with elementary functions, the Risch algorithm is far from elementary; {{as of|2025|lc=y}}, it seems that no complete implementation is available.
+[[Liouville's theorem (differential algebra)|Liouville's result]] is that, if an elementary function has an elementary antiderivative, then this antiderivative is a linear combination of logarithms, where the coefficients and the arguments of the logarithms are elementary functions involved, in some sense, in the definition of the function. The [[Risch algorithm]] (1968) can decide whether an elementary function has an elementary antiderivative, and, if so, to compute it. However, {{As of|2025|lc=y}}, there is no full implementation.<ref>{{Cite web |date=Oct 15, 2020|title=integration - Does there exist a complete implementation of the Risch algorithm?|url=https://mathoverflow.net/questions/374089/does-there-exist-a-complete-implementation-of-the-risch-algorithm|access-date=2023-02-10|website=MathOverflow|language=en}}</ref>
-In late-nineteenth-century analysis, elementary functions were often classified into successive kinds according to the number of independent integrations required for their definition.  Functions expressible without any integration—those generated from rational functions by algebraic operations together with exponentiation, logarithms, and circular or hyperbolic trigonometric functions—were said to be elementary functions of the first kind (in the sense of Liouville).  Functions defined by a single integration of an algebraic function, such as the error function and the elliptic integrals, were elementary functions of the second kind; their inverses, the elliptic functions, were considered of the same order.  Higher "kinds" (third, fourth, etc.) corresponded to multiple integrals of algebraic functions, giving rise to hyperelliptic and more general Abelian functions.{{sfn|Forsyth|1893}}
-The essential point of the classification was that the class of elementary functions of any given kind be closed under the elementary operations—addition, multiplication, composition, and differentiation—so that differentiation never leads outside the same class, while integration may ascend to the next higher kind.
 == Examples ==
 === Basic examples ===
-Elementary functions of a single variable {{mvar|x}} include:
+Elementary functions of a single variable {{tmath|x}} include:
-* [[Constant function]]s: <math>2,\ \pi,\ e,</math> the [[Euler–Mascheroni constant]], [[Apéry's constant]], [[Khinchin's constant]], etc.  Any constant real (or complex) number.
+* [[Constant function]]s: {{tmath|2}}, {{tmath|\pi}}, {{tmath|e}}, the [[Euler–Mascheroni constant]], [[Apéry's constant]], [[Khinchin's constant]], etc.  Any constant real (or complex) number.
-* [[Exponentiation|Powers of {{tmath|x}}]]: <math>x^\alpha=e^{\alpha\log x}</math> etc. (The exponent can be any real or complex constant.)
+* [[Exponentiation|Powers of {{tmath|x}}]]: {{tmath|1= x^\alpha=e^{\alpha\log x} }}, etc. (The exponent can be any real or complex constant.)
-* [[Exponential function]]s: <math>\textstyle e^x,\quad a^x=e^{x\log a}</math>
+* [[Exponential function]]s: {{tmath|e^x}}, {{tmath|1=\textstyle a^x=e^{x\log a} }}
-* [[Logarithm]]s: <math>\textstyle \log x, \quad\log_a x=\frac {\log x}{\log a}</math>
+* [[Logarithm]]s: {{tmath|\log x}}, {{tmath|1=\textstyle \log_a x=\frac {\log x}{\log a} }}
-* [[Trigonometric function]]s: <math>\textstyle\sin x=\frac{e^{ix}-e^{-ix}}{2i},\ \cos x=\frac{e^{ix}+e^{-ix}}{2},\ \tan x=\frac{\sin x}{\cos x},\ </math> etc.
+* [[Trigonometric function]]s: {{tmath|1=\textstyle \sin x=\frac{e^{ix}-e^{-ix} }{2i} }}, {{tmath|1=\textstyle \cos x=\frac{e^{ix}+e^{-ix} }{2} }}, {{tmath|1=\textstyle \tan x=\frac{\sin x}{\cos x} }}, etc.
-* [[Inverse trigonometric function]]s: <math>\arcsin x,\ \arccos x,</math> etc.
+* [[Inverse trigonometric function]]s: {{tmath|\arcsin x}}, {{tmath|\arccos x}}, etc.
-* [[Hyperbolic function]]s: <math>\sinh x,\ \cosh x,</math> etc.
+* [[Hyperbolic function]]s: {{tmath|\sinh x}}, {{tmath|\cosh x}}, etc.
-* [[Inverse hyperbolic function]]s: <math>\operatorname{arsinh} x,\ \operatorname{arcosh} x,</math> etc.
+* [[Inverse hyperbolic function]]s: {{tmath|\operatorname{arsinh} x}}, {{tmath|\operatorname{arcosh} x}}, etc.
-* All functions obtained by adding, subtracting, multiplying or dividing a finite number of any of the previous functions<ref>{{cite book|author=Morris Tenenbaum|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 adding, subtracting, multiplying or dividing a finite number of any of the previous functions<ref>{{cite book |author=Morris Tenenbaum |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 as [[root of a polynomial|roots]] of a polynomial whose coefficients are elementary functions<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><ref>Ritt, chapter 1</ref>
+* All functions obtained as [[root of a polynomial|roots]] of a polynomial whose coefficients are elementary functions<ref name=":1">{{cite book |title=Calculus |last=Spivak, Michael |author-link=Michael Spivak |date=1994 |publisher=Publish or Perish |isbn=0914098896 |edition=3rd |location=Houston, Tex. |page=363 |oclc=31441929 }}</ref><ref>Ritt, chapter 1</ref>
 * 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.
+Certain elementary functions of a single complex variable {{tmath|z}}, such as <math>\sqrt{z}</math> and {{tmath|\log z}}, 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)
+* Addition, e.g. ({{tmath|x + 1}})
-* Multiplication, e.g. (2{{mvar|x}})
+* Multiplication, e.g. ({{tmath|2x}})
-*[[Polynomial]] functions
+* [[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>
-*<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]].
+The last function is equal to {{tmath|\arccos x}}, 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.
 === Non-elementary functions ===
-All elementary functions are [[Analytic function|analytic]] in the following sense: they can be extended to [[functions of a complex variable]] (possibly [[multivalued function|multivalued]]) that are analytic except at finitely many points of the [[complex plane]].<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> Thus nonanalytic functions such as the [[absolute value]] function are not elementary,<ref>Watson and Whittaker 1927, footnote to p 82. In the context of elementary functions, the function <math>y=f(x)</math> defined as the root of <math>y^2-x^2=0</math> is two-valued: <math>y=\pm x</math>.</ref> nor are most other [[piecewise-defined function]]s.
+All elementary functions are [[Analytic function|analytic]] in the following sense: they can be extended to [[functions of a complex variable]] (possibly [[multivalued function|multivalued]]) that are analytic except at isolated points of the [[complex plane]].<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> Thus nonanalytic functions such as the [[absolute value]] function are not elementary,<ref>Watson and Whittaker 1927, footnote to p 82. In the context of elementary functions, the function <math>y=f(x)</math> defined as the root of <math>y^2-x^2=0</math> is two-valued: <math>y=\pm x</math>.</ref> nor are most other [[piecewise-defined function]]s.
 Not every analytic function is elementary. In fact, most [[special function]]s are not elementary. Non-elementary functions include:
@@ Line 49: / Line 44: @@
 * 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]]
+** the [[error function]], {{tmath|1= \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 [[Risch algorithm]]
-* other [[nonelementary integral]]s, including the [[Dirichlet integral]] and [[elliptic integral]]
+* other [[nonelementary integral]]s, including the [[Dirichlet integral]] and [[elliptic integral]].
+== Real-variables and analytic branches ==
+In elementary real-variable settings such as those in calculus and pre-calculus, expressions involving roots, logarithms, and inverse trigonometric functions are often interpreted using fixed real branches on specified real domains. This convention is distinct from the analytic convention used in the theory of elementary functions and integration in finite terms. Risch gives a precise definition of elementary functions "in the sense of analysis" by using functions of a complex variable rather than a real variable. In this setting the elementary functions are built using algebraic operations, exponentials, and logarithms, and are represented in differential fields of meromorphic functions on regions of the complex plane or on Riemann surfaces.{{sfn|Risch|1969|pp=167–168}}
+An algebraic equation such as
+<math display="block">y^2=x^2</math>
+has the local analytic branches <math>y=x</math> and <math>y=-x</math>.  The real identity
+<math display="block">\sqrt{x^2}=|x|</math>
+uses the convention that <math>\sqrt{\cdot}</math> denotes the nonnegative real square root, and so changes from one analytic branch to the other at <math>x=0</math>. Thus the restrictions of <math>|x|</math> to <math>(0,\infty)</math> and <math>(-\infty,0)</math> are elementary, but the usual real absolute value function on an interval containing <math>0</math> is not a single analytic branch.
+The same distinction appears in complex analysis.  Whittaker and Watson note that although <math>x-iy</math> and <math>|z|</math>, where <math>z=x+iy</math>, are functions of <math>z</math> in a general sense, they are not elementary functions of the analytic type under consideration.{{sfn|Whittaker|Watson|1927|loc=§5.1}}  In symbolic computation, functions such as absolute value, signum, and piecewise-defined functions can be treated instead by adjoining a step or conditional operation, which forms a separate class of piecewise function rings.{{sfn|von Mohrenschildt|1998}}
 == Closure ==
 It follows directly from the definition that the set of elementary functions is [[closure (mathematics)|closed]] under arithmetic operations, (algebraic) 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==
+== Extensions ==
-Some have proposed extending the set of elementary functions by extending with certain [[transcendental function]]s, 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> or [[elliptic function]]s,<ref>Ince, E. L. (1956) [1926]. ''Ordinary Differential Equations''. New York: Dover Publications. ISBN 0-486-60339-4, footnote to p 330</ref> all of which are analytic. The key attribute, from the perspective of the Liouville theorem, is that as a class, they are closed under taking derivatives. For example, the Lambert function <math>w=W(z)</math>, which is defined implicitly by the equation <math>we^w=z</math>, has a derivative which can be obtained by [[implicit differentiation]]:
+In late-nineteenth-century analysis, elementary functions were often classified into successive kinds according to the number of independent integrations required for their definition.  Functions expressible without any integration—those generated from rational functions by algebraic operations together with exponentiation, logarithms, and circular or hyperbolic trigonometric functions—were said to be elementary functions of the first kind (in the sense of Liouville).  Functions defined by a single integration of an algebraic function, such as the error function and the elliptic integrals, were elementary functions of the second kind; their inverses, the elliptic functions, were considered of the same order.  Higher "kinds" (third, fourth, etc.) corresponded to multiple integrals of algebraic functions, giving rise to hyperelliptic and more general Abelian functions.{{sfn|Forsyth|1893}}
-<math>W'(z) = \frac{e^{-W(z)}}{1+W(z)},</math>
+The essential point of the classification was that the class of elementary functions of any given kind be closed under the elementary operations—addition, multiplication, composition, and differentiation—so that differentiation never leads outside the same class, while integration may ascend to the next higher kind.
+More recently, some have proposed extending the set of elementary functions by extending with certain [[transcendental function]]s, 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> or [[elliptic function]]s,<ref>Ince, E. L. (1956) [1926]. ''Ordinary Differential Equations''. New York: Dover Publications. ISBN 0-486-60339-4, footnote to p 330</ref> all of which are analytic. The key attribute, from the perspective of the Liouville theorem, is that as a class, they are closed under taking derivatives. For example, the Lambert function <math>w=W(z)</math>, which is defined implicitly by the equation {{tmath|1= we^w=z }}, has a derivative which can be obtained by [[implicit differentiation]]:
+: <math>W'(z) = \frac{e^{-W(z)}}{1+W(z)},</math>
 which is again "elementary", provided that <math>W(z)</math> is.
+== Differential algebra ==
 The mathematical definition of an ''elementary function'' is formalized in [[differential algebra]].  A [[differential field]] is a [[field (mathematics)|field]]  with an 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'' {{tmath|F}} is a field  together with a [[derivation (differential algebra)|derivation]]  {{tmath|u\mapsto \partial u}} that maps {{tmath|F}}  to itself. The derivation generalizes [[derivative]], being linear (thaat is, {{tmath|1=\partial (u + v) = \partial u + \partial v}}) and satisfying the [[product rule|Leibniz product rule]] (that is,{{tmath|1=\partial(u\cdot v)=\partial u\cdot v+u\cdot\partial v}}) for every two elements {{tmath|u}} and {{tmath|v}} in {{tmath|F}}. The [[rational function]]s over {{tmath|\Q}} of {{tmath|\C}} form a basic examples of differential fields, when equipped with the usual derivative.
+A ''differential field'' {{tmath|F}} is a field  together with a [[derivation (differential algebra)|derivation]]  {{tmath|u\mapsto \partial u}} that maps {{tmath|F}}  to itself. The derivation generalizes [[derivative]], being linear (that is, {{tmath|1=\partial (u + v) = \partial u + \partial v}}) and satisfying the [[product rule|Leibniz product rule]] (that is,{{tmath|1=\partial(u\cdot v)=\partial u\cdot v+u\cdot\partial v}}) for every two elements {{tmath|u}} and {{tmath|v}} in {{tmath|F}}. The [[rational function]]s over {{tmath|\Q}} of {{tmath|\C}} form a basic examples of differential fields, when equipped with the usual derivative.
-An element {{math|''h''}} of {{tmath|F}} is a constant if {{tmath|1=\partial h=0}}. The constants of {{tmath|F}} form a dfferential field with zero derivative. Care must be taken that a differential field extension of a differential field may enlarge the field of constants.
+An element {{tmath|h}} of {{tmath|F}} is a constant if {{tmath|1=\partial h=0}}. The constants of {{tmath|F}} form a differential field with zero derivative. Care must be taken that a differential field extension of a differential field may enlarge the field of constants.
-A function {{mvar|u}} of a differential extension {{mvar|G}} of a differential field {{mvar|F}} is an '''elementary function''' over {{mvar|F}} if it belongs to a finite chain (for inclusion) of differential subfields of {{mvar|G}} that starts from {{mvar|F}} and is such that each is generated over the preceding one by a function that is either
+A function {{tmath|u}} of a differential extension {{tmath|G}} of a differential field {{tmath|F}} is an '''elementary function''' over {{tmath|F}} if it belongs to a finite chain (for inclusion) of differential subfields of {{tmath|G}} that starts from {{tmath|F}} and is such that each is generated over the preceding one by a function that is either
 * [[Algebraic function|algebraic]] over the preceding field, or
-* an ''exponential'', that is, {{tmath|1=\partial u = u\partial a}}  for some {{tmath|a\in F}}, or
+* an ''exponential'', that is, {{tmath|1=\partial u = u\partial a}} for some {{mvar|a}} belonging to a prior subfield, or
-* a ''logarithm'', that is, {{tmath|1=\partial u = \partial a/a}} for some {{tmath|a\in F}}.
+* a ''logarithm'', that is, {{tmath|1=\partial u = \partial a/a}} for some {{mvar|a}} belonging to a prior subfield, (see [[Liouville's theorem (differential algebra)|Liouville's theorem]])
-(see [[Liouville's theorem (differential algebra)|Liouville's theorem]])
 With this definition, the usual elementary functions are exactly the function that are elementary over the field of the [[rational function]]s. This generalized definition allows considering every transcendental function as elementary for applying Liouville's theorem.
-==See also==
+== See also ==
 * [[Algebraic function]]
@@ Line 81: / Line 92: @@
 * {{anl|Elementary function arithmetic}}
 * {{anl|Liouville's theorem (differential algebra)}}
+* [[Richardson's theorem]]: Undecidability of inequalities of real numbers formed out of a class of elementary functions
 * [[Tarski's high school algebra problem]]
 * {{anl|Transcendental function}}
-==Notes==
+== Notes ==
 {{reflist}}
-==References==
+== References ==
 * {{cite book |last=Forsyth |first=Andrew |author-link=Andrew Forsyth |title=Theory of Functions of a Complex Variable |date=1893 |url=https://archive.org/details/theoryoffunction00fors/?q=region |publisher=Cambridge |jfm=25.0652.01 }}
-*{{Cite journal
+* {{cite journal
   | last = Liouville
   | first = Joseph
@@ Line 100: / Line 112: @@
   | url = http://gallica.bnf.fr/ark:/12148/bpt6k433678n/f127.item.r=Liouville
 }}
-*{{Cite journal
+* {{cite journal
   | last = Liouville
   | first = Joseph
@@ Line 111: / Line 123: @@
   | url = http://gallica.bnf.fr/ark:/12148/bpt6k433678n/f152.item.r=Liouville
 }}
-*{{Cite journal
+* {{cite journal
   | last = Liouville
   | first = Joseph
@@ Line 122: / Line 134: @@
   | url = http://gdz.sub.uni-goettingen.de/en/dms/loader/img/?PID=GDZPPN002139332
 }}
-*{{Cite book
+* {{Citation
+ |last=von Mohrenschildt
+ |first=Martin
+ |title=A Normal Form for Function Rings of Piecewise Functions
+ |journal=Journal of Symbolic Computation
+ |volume=26
+ |issue=5
+ |year=1998
+ |pages=607–619
+ |doi=10.1006/jsco.1998.0229
+}}.
+*{{citation
+ | last = Risch
+ | first = R. H.
+ | title = The problem of integration in finite terms
+ | journal = [[Transactions of the American Mathematical Society]]
+ | year = 1969
+ | volume = 139
+ | pages = 167–189
+ | publisher = American Mathematical Society
+ | doi = 10.2307/1995313
+ | jstor = 1995313
+ | doi-access = free
+ }}.
+* {{cite book
   | last = Ritt
   | first = Joseph
@@ Line 131: / Line 167: @@
   | url = https://www.ams.org/online_bks/coll33/
 }}
-*{{Cite journal
+* {{cite journal
   | last = Rosenlicht
   | first = Maxwell
@@ Line 144: / Line 180: @@
   | jstor=2318066
 }}
+*{{cite book |title=A Course Of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions |author-last1=Whittaker |author-first1=Edmund Taylor |author-link1=Edmund Taylor Whittaker |author-last2=Watson |author-first2=George Neville |author-link2=George Neville Watson |date=1927-01-02 |edition=4th |publisher=[[at the University Press]] |publication-place=Cambridge, UK}}
-==Further reading==
+== 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==
+== External links ==
 * [https://www.encyclopediaofmath.org/index.php/Elementary_functions ''Elementary functions'' at Encyclopaedia of Mathematics]
 * {{MathWorld|ElementaryFunction|Elementary function}}
-{{Authority control}}
+{{authority control}}
 {{DEFAULTSORT:Elementary Function}}
 [[Category:Differential algebra]]
 [[Category:Computer algebra]]
 [[Category:Types of functions]]

Elementary function: Difference between revisions

Latest revision as of 00:00, 30 June 2026

Contents

Examples

Basic examples

Composite examples

Non-elementary functions

Real-variables and analytic branches

Closure

Extensions

Differential algebra

See also

Notes

References

Further reading

External links

Navigation menu

Elementary function: Difference between revisions

Latest revision as of 00:00, 30 June 2026

Examples

Basic examples

Composite examples

Non-elementary functions

Real-variables and analytic branches

Closure

Extensions

Differential algebra

See also

Notes

References

Further reading

External links

Navigation menu

Search