PlanetPhysics/Elementary Function

An elementary function is a real [[../Bijective/|function]] (of one variable) that can be constructed by a finite number of elementary [[../Cod/|operations]] (addition, subtraction, multiplication and division) and [[../Cod/|compositions]] from constant functions, the [[../Cod/|identity]] function ($x\mapsto x$), [[../CoIntersections/|algebraic]] functions, exponential functions, logarithm functions, trigonometric functions and cyclometric functions.

Examples

• Consequently, the polynomial functions, the absolute value\, $|x|=\sqrt{x^2}$,\, the triangular-wave function\, $\arcsin (\sin x)$, the [[../PowerFunction/|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\, $x>0$).
• $\zeta (x):=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^x}$\, and\, $\mathrm{Li}x:=\underset{2}{\overset{x}{\int }}\frac{dt}{\ln t}$\, are not elementary functions --- it may be shown that they can not be expressed is such a way which is required in the definition.