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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 $\arcsin(\sin{x})$, 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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