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In mathematics, an **elementary function** is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponential functions, including possibly their inverse functions (e.g., arcsin, log, or *x*^{1/n}).^{[1]}

All elementary functions are continuous on their domains.

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]}

Elementary functions of a single variable x include:

- Constant functions: etc.
- Rational powers of x: etc.
- more general algebraic functions: satisfying , which is not expressible through n-th roots or rational powers of x alone
- Exponential functions:
- Logarithms:
- Trigonometric functions: etc.
- Inverse trigonometric functions: etc.
- Hyperbolic functions: etc.
- Inverse hyperbolic functions: etc.
- All functions obtained by adding, subtracting, multiplying or dividing a finite number of any of the previous functions
^{[6]} - All functions obtained by root extraction of a polynomial with coefficients in elementary functions
^{[7]} - 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 and , may be multivalued. Additionally, certain classes of functions may be obtained by others using the final two rules. For example, the exponential function composed with addition, subtraction, and division provides the hyperbolic functions, while initial composition with instead provides the trigonometric functions.

Examples of elementary functions include:

- Addition, e.g. (x+1)
- Multiplication, e.g. (2x)
- Polynomial functions

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

All monomials, polynomials, rational functions and algebraic functions are elementary. The absolute value function, for real , is also elementary as it can be expressed as the composition of a power and root of : .

An example of a function that is *not* elementary is the error function

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.

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

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 *F*_{0} (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

and satisfies the Leibniz product rule

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)

- Algebraic function – Function that can be defined as the root of a polynomial equation
- Closed-form expression – Mathematical formula involving a given set of operations
- Differential Galois theory – Study of Galois symmetry groups of differential fields
- Elementary function arithmetic – System of arithmetic in proof theory
- Liouville's theorem (differential algebra) – Says when antiderivatives of elementary functions can expressed as elementary functions
- Tarski's high school algebra problem – Mathematical problem
- Transcendental function – Analytic function that does not satisfy a polynomial equation
- Tupper's self-referential formula – Formula that visually represents itself when graphed

**^**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.**^**Weisstein, Eric W. "Elementary Function." From MathWorld

- 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.

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