Elementary function

In mathematics, an elementary function is a function of a single variable composed of particular simple functions.

Elementary functions are typically defined as a sum, product, and/or composition of finitely many polynomials, rational functions, trigonometric and exponential functions, and their inverse functions (including arcsin, log, x1/n).[1]

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

The elementary functions (of x) include:

Constant functions: \( {\displaystyle 2,\ \pi ,\ e,} \)etc.
Powers of x: \( {\displaystyle x,\ x^{2},\ x^{3},} \) etc.
Roots of \( {\displaystyle x{\text{: }}{\sqrt {x}},\ {\sqrt[{3}]{x}},} \) etc.
Exponential functions: \( e^{x} \)
Logarithms: \( \log x \)
Trigonometric functions: \( {\displaystyle \sin x,\ \cos x,\ \tan x,} \) etc.
Inverse trigonometric functions: \( {\displaystyle \arcsin x,\ \arccos x,} \)etc.
Hyperbolic functions: \( {\displaystyle \sinh x,\ \cosh x,} \)etc.
Inverse hyperbolic functions: \( {\displaystyle \operatorname {arsinh} x,\ \operatorname {arcosh} x,} \) etc.
All functions obtained by adding, subtracting, multiplying or dividing any of the previous functions[6]
All functions obtained by composing previously listed functions

Some elementary functions, such as roots, logarithms, or inverse trigonometric functions, are not entire functions and may be multivalued.
Composite examples

Examples of elementary functions include:

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

The last function is equal to \( {\displaystyle \arccos x}, \) the inverse cosine, in the entire complex plane.

All monomials, polynomials and rational functions are elementary. Also, 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: \( |x|={\sqrt {x^{2}}}. \)
Non-elementary functions

An example of a function that is not elementary is the error 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 the Risch algorithm.

See also the examples in Liouvillian function and Nonelementary integral.

Closure

It follows directly from the definition that the set of elementary functions is closed under arithmetic operations 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

\( \partial (u+v)=\partial u+\partial v \)

and satisfies the Leibniz product rule

\( \partial (u\cdot v)=\partial u\cdot v+u\cdot \partial v\,. \)

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

Closed-form expression
Differential Galois theory
Algebraic function
Transcendental function

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.

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.