elementary function
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 triangularwave function $\mathrm{arcsin}(\mathrm{sin}x)$, the power function^{} ${x}^{\pi}={e}^{\pi \mathrm{ln}x}$ and the function ${x}^{x}={e}^{x\mathrm{ln}x}$ are elementary functions (N.B., the real power functions entail that $x>0$).

•
$\zeta (x):={\displaystyle \sum _{n=1}^{\mathrm{\infty}}}{\displaystyle \frac{1}{{n}^{x}}}$ and $\mathrm{Li}x:={\displaystyle {\int}_{2}^{x}}{\displaystyle \frac{dt}{\mathrm{ln}t}}$ are not elementary functions — it may be shown that they can not be expressed is such a way which is required in the definition.
Title  elementary function 

Canonical name  ElementaryFunction 
Date of creation  20130322 14:46:29 
Last modified on  20130322 14:46:29 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  18 
Author  pahio (2872) 
Entry type  Definition 
Classification  msc 26A99 
Related topic  RiemannZetaFunction 
Related topic  LogarithmicIntegral 
Related topic  AlgebraicFunction 
Related topic  TableOfMittagLefflerPartialFractionExpansions 