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

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

单词	ElementaryFunction
释义	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 $\\displaystyle\\operatorname{Li}{x}:=\\int_{2}^{x}\\frac{dt}{\\ln{t}}$ are not elementary functions — it may be shown that they can not be expressed is such a way which is required in the definition.
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