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Concept
Elementary function
Summary
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 x1/n).
All elementary functions are continuous on their domains.
Elementary functions were introduced by Joseph Liouville in a series of papers from 1833 to 1841. An algebraic treatment of elementary functions was started by Joseph Fels Ritt in the 1930s.
Elementary functions of a single variable x include:
Constant functions: etc.
Rational powers of x: etc.
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
All functions obtained by root extraction of a polynomial with coefficients in 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 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 : .
Official source
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