Elementary function

Template:Short description 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 the absolute value function and piecewise-defined functions. More generally, in modern mathematics, elementary functions comprise the set of functions previously enumerated, all algebraic functions (not often encountered by beginners), 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 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. 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; Template:As of, 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.Template:Sfn

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 Template:Mvar include:

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

Certain elementary functions of a single complex variable Template:Mvar, such as $\sqrt{z}$ and $\log z$, 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.

Composite examples

Examples of elementary functions include:

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

The last function is equal to $\arccos x$, 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 finitely many points of the complex plane. Thus nonanalytic functions such as the absolute value function are not elementary, 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, $\mathrm{e}\mathrm{r}\mathrm{f}(x)=\frac{2}{\sqrt{\pi }}\underset{0}{\overset{x}{\int }}{e}^{-t^2}dt,$ 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

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.

Differential algebra

Some have proposed extending the set of elementary functions by extending with certain transcendental functions, to include, for example, the Lambert W function or elliptic functions, 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 $w{e}^w=z$, has a derivative which can be obtained by implicit differentiation: $W^{\prime }(z)=\frac{e^{-W(z)}}{1+W(z)},$ which is again "elementary", provided that $W(z)$ is.

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 (thaat 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:Math of Template:Tmath is a constant if Template:Tmath. The constants of Template:Tmath 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 function Template:Mvar of a differential extension Template:Mvar of a differential field Template:Mvar is an elementary function over Template:Mvar if it belongs to a finite chain (for inclusion) of differential subfields of Template:Mvar that starts from Template:Mvar 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:Tmath, or
• a logarithm, that is, Template:Tmath for some Template:Tmath.

(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.

See also

• Algebraic function
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• Differential Galois theory
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• Tarski's high school algebra problem
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External links

• Elementary functions at Encyclopaedia of Mathematics