Movatterモバイル変換

[0]ホーム

Jump to content

Elementary function

Edit links

From Wikipedia, the free encyclopedia

Type of mathematical function

Not to be confused withElementary recursive function.

Inmathematics, anelementary function is afunction of a singlevariable (real orcomplex) that is typically encountered by beginners. The basic elementary functions arepolynomial functions,rational functions, thetrigonometric functions, theexponential andlogarithm functions, then-th root, and theinverse trigonometric functions, as well as those functions obtained byaddition,multiplication,division, andcomposition of these. Some functions which are encountered by beginners arenot elementary, such as theabsolute value function andpiecewise-defined functions. More generally, in modern mathematics, elementary functions comprise the set of functions previously enumerated, allalgebraic functions (not often encountered by beginners), and all functions obtained byroots of a polynomial whose coefficients are elementary.

This list of elementary functions was originally set forth byJoseph Liouville in 1833. A key property is that all elementary functions havederivatives of any order, which are also elementary, and can bealgorithmically computed by applying thedifferentiation rules (or the rules forimplicit differentiation in the case of roots). TheTaylor series of an elementary function converges in a neighborhood of every point of its domain. More generally, they areglobal analytic functions, defined (possibly withmultiple values, such as the elementary function ${\sqrt {z}}$ or $\log z$ ) for everycomplex argument, except atisolated 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. More than 130 years later,Risch algorithm, named afterRobert 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^[update], it seems that no complete implementation is available.

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.^[1]

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

[edit]

Basic examples

[edit]

Elementary functions of a single variablex include:

Constant functions: $2,\ \pi ,\ e,$ theEuler–Mascheroni constant,Apéry's constant,Khinchin's constant, etc. Any constant real (or complex) number.
Powers of⁠ $x {\displaystyle x}$ ⁠: $x^{\alpha }=e^{\alpha \log x}$ etc. (The exponent can be any real or complex constant.)
Exponential functions: $\textstyle e^{x},\quad a^{x}=e^{x\log a}$
Logarithms: $\textstyle \log x,\quad \log _{a}x={\frac {\log x}{\log a}}$
Trigonometric functions: $\textstyle \sin x={\frac {e^{ix}-e^{-ix}}{2i}},\ \cos x={\frac {e^{ix}+e^{-ix}}{2}},\ \tan x={\frac {\sin x}{\cos x}},\$ etc.
Inverse trigonometric functions: $\arcsin x,\ \arccos x,$ etc.
Hyperbolic functions: $\sinh x,\ \cosh x,$ etc.
Inverse hyperbolic functions: $\operatorname {arsinh} x,\ \operatorname {arcosh} x,$ etc.
All functions obtained by adding, subtracting, multiplying or dividing a finite number of any of the previous functions^[2]
All functions obtained asroots of a polynomial whose coefficients are elementary functions^[3]^[4]
All functions obtained bycomposing a finite number of any of the previously listed functions

Certain elementary functions of a single complex variablez, such as ${\sqrt {z}}$ and $\log z$ , may bemultivalued. 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 $i z {\displaystyle iz}$ instead provides the trigonometric functions.

Composite examples

[edit]

Examples of elementary functions include:

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

The last function is equal to $\arccos x$ , theinverse cosine, in the entirecomplex plane.

Allmonomials,polynomials,rational functions andalgebraic functions are elementary.

Non-elementary functions

[edit]

All elementary functions areanalytic in the following sense: they can be extended tofunctions of a complex variable (possiblymultivalued) that are analytic except at finitely many points of thecomplex plane.^[5] Thus nonanalytic functions such as theabsolute value function are not elementary,^[6] nor are most otherpiecewise-defined functions.

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

thegamma function
non-elementaryLiouvillian functions, including
- theexponential integral (Ei)logarithmic integral (Li orli) andFresnel integrals (S andC)
- theerror function, $\mathrm {erf} (x)={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}\,dt,$ a fact that may not be immediately obvious, but can be proven using theRisch algorithm
othernonelementary integrals, including theDirichlet integral andelliptic integral

Closure

[edit]

It follows directly from the definition that the set of elementary functions isclosed under arithmetic operations, (algebraic) root extraction and composition. The elementary functions are closed underdifferentiation. They are not closed underlimits and infinite sums. Importantly, the elementary functions arenot closed underintegration, as shown byLiouville's theorem, seenonelementary integral. TheLiouvillian functions are defined as the elementary functions and, recursively, the integrals of the Liouvillian functions.

Differential algebra

[edit]

Some have proposed extending the set of elementary functions by extending with certaintranscendental functions, to include, for example, theLambert W function^[7] orelliptic 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 $we^{w}=z$ , has a derivative which can be obtained byimplicit differentiation: $W'(z)={\frac {e^{-W(z)}}{1+W(z)}},$ which is again "elementary", provided that $W(z)$ is.

The mathematical definition of anelementary function is formalized indifferential algebra. Adifferential field is afield with an extra operation of derivation (algebraic version of differentiation). Using the derivation operation new equations can be written and their solutions used inextensions of the algebra. By starting with thefield ofrational functions, two special types of transcendental extensions (the logarithm and the exponential) can be added to the field building a tower containing elementary functions.

Adifferential field⁠ $F {\displaystyle F}$ ⁠ is a field together with aderivation⁠ $u\mapsto \partial u$ ⁠ that maps⁠ $F {\displaystyle F}$ ⁠ to itself. The derivation generalizesderivative, being linear (thaat is,⁠ $\partial (u+v)=\partial u+\partial v$ ⁠) and satisfying theLeibniz product rule (that is,⁠ $\partial (u\cdot v)=\partial u\cdot v+u\cdot \partial v$ ⁠) for every two elements⁠ $u {\displaystyle u}$ ⁠ and⁠ $v {\displaystyle v}$ ⁠ in⁠ $F {\displaystyle F}$ ⁠. Therational functions over⁠ $\mathbb {Q}$ ⁠ of⁠ $\mathbb {C}$ ⁠ form a basic examples of differential fields, when equipped with the usual derivative.

An elementh of⁠ $F {\displaystyle F}$ ⁠ is a constant if⁠ $\partial h=0$ ⁠. The constants of⁠ $F {\displaystyle F}$ ⁠ 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.

A functionu of a differential extensionG of a differential fieldF is anelementary function overF if it belongs to a finite chain (for inclusion) of differential subfields ofG that starts fromF and is such that each is generated over the preceding one by a function that is either

algebraic over the preceding field, or
anexponential, that is,⁠ $\partial u=u\partial a$ ⁠ for some⁠ $a\in F$ ⁠, or
alogarithm, that is,⁠ $\partial u=\partial a/a$ ⁠ for some⁠ $a\in F$ ⁠.

(seeLiouville's theorem)

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