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 $x > 0$ ).
• $\displaystyle\zeta(x) := \sum_{n = 1}^{\infty}\frac{1}{n^x}$ 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.