PlanetPhysics/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 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\, $$x > 0$$).
 * $$\zeta(x) := \sum_{n = 1}^{\infty}\frac{1}{n^x}$$\, and\, $$\operatorname{Li}{x} := \int_2^{x}\frac{dt}{\ln{t}}$$\, are not elementary functions --- it may be shown that they can not be expressed is such a way which is required in the definition.