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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  $x > 0$).
  • $\displaystyle\zeta(x) := \sum_{n = 1}^{\infty}\frac{1}{n^x}$  and  $\displaystyle\operatorname{Li}{x} := \int_2^{x}\frac{dt}{\ln{t}}$  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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Cross-references: power function, algebraic, identity, compositions, operations, function

This is version 1 of elementary function, born on 2009-04-18.
Object id is 658, canonical name is ElementaryFunction.
Accessed 279 times total.

Classification:
Physics Classification: 02.30.-f (Function theory, analysis)
Pending Errata and Addenda
None.
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