Elementary function

Not to be confused with Elementary recursive function.

In mathematics, elementary functions are those functions that are most commonly encountered by beginners. They are typically real functions of a single real variable that can be defined by applying the operations of addition, multiplication, division, nth root, and function composition to polynomial, exponential, logarithm, and trigonometric functions. They include inverse trigonometric functions, hyperbolic functions and inverse hyperbolic functions, which can be expressed in terms of logarithms and exponential function.

All elementary functions have derivatives of any order, which are also elementary, and can be algorithmically computed by applying the differentiation rules. 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 ${\displaystyle {\sqrt {z}}}$ or ${\displaystyle \log z}$) 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.

In an attempt to solve this problem, Joseph Liouville introduced in 1833 a definition of elementary functions that extends the above one and is commonly accepted:^[1][2][3] An elementary function is a function that can be built, using addition, multiplication, division, and function composition, from constant functions, exponential functions, the complex logarithm, and roots of polynomials with elementary functions as coefficients. This includes the trigonometric functions, since, for example, ⁠${\displaystyle \textstyle \cos x={\frac {e^{ix}+e^{-ix}}{2}}}$⁠, as well as every algebraic function.

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^[update], it seems that no complete implementation is available.

Examples

Basic examples

Elementary functions of a single variable x include:

• Constant functions: ${\displaystyle 2,\ \pi ,\ e,}$  the Euler–Mascheroni constant, Apéry's constant, Khinchin's constant, etc. Any constant real (or complex) number.
• Powers of ⁠${\displaystyle x}$ ⁠: ${\displaystyle x^{\alpha }=e^{\alpha \log x}}$  etc. (The exponent can be any real or complex constant.)
• Exponential functions: ${\displaystyle \textstyle e^{x},\quad a^{x}=e^{x\log a}}$ 
• Logarithms: ${\displaystyle \textstyle \log x,\quad \log _{a}x={\frac {\log x}{\log a}}}$ 
• Trigonometric functions: ${\displaystyle \textstyle \sin x={\frac {e^{ix}-e^{-ix}}{2i}},\ \cos x={\frac {e^{ix}+e^{-ix}}{2}},\ \tan x={\frac {\sin x}{\cos x}},\ }$  etc.
• Inverse trigonometric functions: ${\displaystyle \arcsin x,\ \arccos x,}$  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 a finite number of any of the previous functions^[4]
• All functions obtained as roots of a polynomial whose coefficients are elementary functions^[5][6]
• 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 ${\displaystyle {\sqrt {z}}}$  and ${\displaystyle \log z}$ , may be multivalued. Additionally, certain classes of functions may be obtained by others using the final two rules. For example, the exponential function ${\displaystyle e^{z}}$  composed with addition, subtraction, and division provides the hyperbolic functions, while initial composition with ${\displaystyle iz}$  instead provides the trigonometric functions.

Composite examples

Examples of elementary functions include:

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

The last function is equal to ${\displaystyle \arccos x}$ , 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.^[7] Thus nonanalytic functions such as the absolute value function are not elementary,^[8] 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, ${\displaystyle \mathrm {erf} (x)={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}\,dt,}$  a fact that may not be immediately obvious, but can be proven using the Risch algorithm
• other nonelementary 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 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

Some have proposed extending the set of elementary functions by extending with certain transcendental functions, to include, for example, the Lambert W function^[9] or elliptic functions,^[10] 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 ${\displaystyle w=W(z)}$ , which is defined implicitly by the equation ${\displaystyle we^{w}=z}$ , has a derivative which can be obtained by implicit differentiation: ${\displaystyle W'(z)={\frac {e^{-W(z)}}{1+W(z)}},}$  which is again "elementary", provided that ${\displaystyle W(z)}$  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 ⁠${\displaystyle F}$ ⁠ is a field together with a derivation ⁠${\displaystyle u\mapsto \partial u}$ ⁠ that maps ⁠${\displaystyle F}$ ⁠ to itself. The derivation generalizes derivative, being linear (thaat is, ⁠${\displaystyle \partial (u+v)=\partial u+\partial v}$ ⁠) and satisfying the Leibniz product rule (that is,⁠${\displaystyle \partial (u\cdot v)=\partial u\cdot v+u\cdot \partial v}$ ⁠) for every two elements ⁠${\displaystyle u}$ ⁠ and ⁠${\displaystyle v}$ ⁠ in ⁠${\displaystyle F}$ ⁠. The rational functions over ⁠${\displaystyle \mathbb {Q} }$ ⁠ of ⁠${\displaystyle \mathbb {C} }$ ⁠ form a basic examples of differential fields, when equipped with the usual derivative.

An element h of ⁠${\displaystyle F}$ ⁠ is a constant if ⁠${\displaystyle \partial h=0}$ ⁠. The constants of ⁠${\displaystyle F}$ ⁠ 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 function u of a differential extension G of a differential field F is an elementary function over F if it belongs to a finite chain (for inclusion) of differential subfields of G that starts from F 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, ⁠${\displaystyle \partial u=u\partial a}$ ⁠ for some ⁠${\displaystyle a\in F}$ ⁠, or
• a logarithm, that is, ⁠${\displaystyle \partial u=\partial a/a}$ ⁠ for some ⁠${\displaystyle a\in F}$ ⁠.

(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
• Closed-form expression
• Differential Galois theory
• Elementary function arithmetic
• Liouville's theorem (differential algebra)
• Tarski's high school algebra problem
• Transcendental function
• Tupper's self-referential formula

Notes

1. Liouville 1833a.
2. Liouville 1833b.
3. Liouville 1833c.
4. Morris Tenenbaum (1985). Ordinary Differential Equations. Dover. p. 17. ISBN 0-486-64940-7.
5. Spivak, Michael. (1994). Calculus (3rd ed.). Houston, Tex.: Publish or Perish. p. 363. ISBN 0914098896. OCLC 31441929.
6. Ritt, chapter 1
7. Risch, Robert H. (1979). "Algebraic Properties of the Elementary Functions of Analysis". American Journal of Mathematics. 101 (4): 743–759. doi:10.2307/2373917. ISSN 0002-9327. JSTOR 2373917.
8. Watson and Whittaker 1927, footnote to p 82. In the context of elementary functions, the function ${\displaystyle y=f(x)}$  defined as the root of ${\displaystyle y^{2}-x^{2}=0}$  is two-valued: ${\displaystyle y=\pm x}$ .
9. Stewart, Seán (2005). "A new elementary function for our curricula?" (PDF). Australian Senior Mathematics Journal. 19 (2): 8–26.
10. Ince, E. L. (1956) [1926]. Ordinary Differential Equations. New York: Dover Publications. ISBN 0-486-60339-4, footnote to p 330

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.

External links

• Elementary functions at Encyclopaedia of Mathematics
• Weisstein, Eric W. "Elementary function". MathWorld.