Talk:PlanetPhysics/Elementary Function

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An {\em elementary function} is a real \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} (of one variable) that can be constructed by a finite number of elementary \htmladdnormallink{operations}{http://planetphysics.us/encyclopedia/Cod.html} (addition, subtraction, multiplication and division) and \htmladdnormallink{compositions}{http://planetphysics.us/encyclopedia/Cod.html} from constant functions, the \htmladdnormallink{identity}{http://planetphysics.us/encyclopedia/Cod.html} function ($x \mapsto x$), \htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html} 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 \htmladdnormallink{power function\,}{http://planetphysics.us/encyclopedia/PowerFunction.html} $x^{\pi} = e^{\pi\ln{x}}$\, 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\, $\displaystyle\operatorname{Li}{x} := \int_2^{x}\frac{dt}{\ln{t}}$\, are not elementary functions --- it may be shown that they can not be expressed is such a way which is required in the definition. \end{itemize}

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