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> the concept of a "function" was undefined but generally understood to be something like what we'd call today an "expression" in a programming language. So, for example, "x^2 + 1" was widely agreed to be a function, but "if x < 0 then x else 0" was controversial Good answer, but not the best example. In many programming languages, the latter is easily written as an expression: (x - abs(x)) / 2
It boils down to what you call an elementary function (https://en.wikipedia.org/wiki/Elementary_function).Taking the absolute of a number generally is not assumed to be in that set, but there is no generally accepted strict definition. A step ‘up’ from elementary functions are special functions (https://en.wikipedia.org/wiki/Special_functions). Likewise, that is loosely defined. For example https://en.wikipedia.org/wiki/List_of_eponyms_of_special_fun... mentions lots of polynomials, one of them https://en.wikipedia.org/wiki/Cyclotomic_polynomial, which definitely are elementary according to https://en.wikipedia.org/wiki/Elementary_function. Wikipedia also contradicts itself in https://en.wikipedia.org/wiki/Closed-form_expression, where it says “a closed form expression or formula is one that is formed with constants, variables, and a set of functions considered as basic and connected by arithmetic operations (+, −, ×, /, and integer powers) and function composition. Commonly, the basic functions that are allowed in closed forms are nth root, exponential function, logarithm, and trigonometric functions” and “For example, if one adds polynomial roots to the basic functions, the functions that have a closed form are called elementary functions” That would put the goniometric functions in the basic set allowed in elementary functions. |