Comment by reikonomusha | Hacker News Reader

Comment by reikonomusha | Hacker News Reader

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The definition of "elementary function" typically includes functions which solve polynomials, like the Bring radical. The definition was developed and is most fitting in algebraic contexts where algebraic structure is meaningful, like Liouvillian structure theorems, algorithmic integration, and computer algebra. See e.g.

- Page 2 and the following example of https://billcookmath.com/courses/math4010-spring2016/math401... (2016)

- Ritt's Integration in Finite Terms: Liouville's Theory of Elementary Methods (1948)

It's not frequent that analysis books will define the class of elementary functions rigorously, but instead refer to examples of them informally.

> See e.g. Page 2 and the following example of https://billcookmath.com/courses/math4010-spring2016/math401... (2016)

There appears to be a typo in that example; I assume "Essentially elementary functions are the functions that can be built from ℂ and f(x) = x" should say something more like "the functions that can be built from ℂ and f(x) = y".

Not a typo! Think of f(x) = x as a seed function that can be used to build other functions. It's one way to avoid talking about "variables" as a "data type" and just keep everything about functions. We can make a function like x + x*exp(log(x)) by "formally" writing

    f + f*(exp∘log)

where + and * are understood to produce new functions. Sort of Haskell-y.

> The definition of "elementary function" typically includes functions which solve polynomials, like the Bring radical.

What. Does that "typical definition" of elementary function includes elliptic functions as well, by any chance?

Not that I've seen.

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