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[parent] 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 $|x| = \sqrt{x^2}$ , 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 $x > 0$ ).
  • $\displaystyle\zeta(x) := \sum_{n = 1}^{\infty}\frac{1}{n^x}$ and $\displaystyle\Li{x} := \int_2^{x}\frac{dt}{\ln{t}}$ 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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See Also: Riemann zeta function, logarithmic integral, algebraic function, table of partial fraction expansions


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function $x^x$ (Example) by pahio
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Cross-references: entail, 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, elementary operations, number, 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 8726 times total.

Classification:
AMS MSC26A99 (Real functions :: Functions of one variable :: Miscellaneous)

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