elementary functionElementaryFunction2004-10-25 15:27:142006-10-14 16:12:45Definitionreal function% this is the default PlanetMath preamble. as your knowledge
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\DeclareMathOperator{\Li}{Li}An {\em 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.
\textbf{Examples}
\begin{itemize}
\item 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\, $x > 0$).
\item $\displaystyle\zeta(x) := \sum_{n = 1}^{\infty}\frac{1}{n^x}$\, 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.
\end{itemize}