PlanetMath: elementary function

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elementary function

(Definition)

An elementary function is a real function (of one variable) that can be constructed by a finite number of elementary operations (addition, subtraction, multiplication and division) and compositions from constant functions, the identity function ( $x \mapsto x$ ), algebraic functions, exponential functions, logarithm functions, trigonometric functions and cyclometric functions.

Examples

Consequently, the polynomial functions, the absolute value $\vert x\vert = \sqrt{x^2}$ , the triangular-wave function $\arcsin(\sin{x})$ , the power function $x^{\pi} = e^{\pi\ln{x}}$ and the function $x^x = e^{x\ln{x}}$ are elementary functions (N.B., the real power functions entail that ).
$\displaystyle\zeta(x) := \sum_{n = 1}^{\infty}\frac{1}{n^x}$ and $\displaystyle{\mathrm{Li}}{x} := \int_2^{x}\frac{dt}{\ln{t}}$ are not elementary functions -- it may be shown that they can not be expressed is such a way which is required in the definition.

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"elementary function" is owned by pahio.

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See Also: Riemann zeta function, logarithmic integral, algebraic function, table of partial fraction expansions

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Attachments:: function (Example) by pahio

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Cross-references: real, function, power function, triangular-wave function, absolute value, polynomial functions, cyclometric functions, trigonometric functions, logarithm functions, exponential functions, algebraic functions, identity function, constant functions, compositions, division, multiplication, subtraction, addition, operations, finite, variable, real function
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This is version 15 of elementary function, born on 2004-10-25, modified 2006-10-14.
Object id is 6420, canonical name is ElementaryFunction.
Accessed 5466 times total.

Classification:

26A99 (Real functions :: Functions of one variable :: Miscellaneous)

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