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Is there a mathematical reason (or possibly a historical one) that the "elementary" functions are what they are? As I'm learning calculus, I seem to focus most of my attention on trigonometric, logarithmic, exponential, and $n$th roots, and solving problems that have solutions which are elementary functions. I've been curious why these functions are called elementary, as opposed to some other functions that turn up rather naturally in mathematics. What is the reason that these functions take up most of our attention, and is there a reason that some additional functions are not included amongst the elementary functions? In other words, what property or properties do these functions possess that separates them from non-elementary functions (if there is one)?

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    thanks for this question. As to what I have posted earlier, there is no elementary function that exists for $x$. Now, I am also wondering about the "distinction" of elementary from those of not2012-03-09
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    Those elementary functions are the familiar functions since from lower level maths. They are the most useful functions regarding mathematics and they play a very vital role in applications. That's makes them elementary.2012-03-09
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    Here is a reverse kind of question/discussion [What is a special function](http://math.stackexchange.com/questions/13067/what-is-a-special-function)2012-03-09
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    The polynomial, exponential, sine, and cosine functions are "elementary" because they are very useful and will more frequently arise naturally in an investigation (whether within math or in an application) than most other functions. So everyone needs to know them. But why are they so useful? I think fundamentally it's because they are solutions to some of the simplest differential equations you could write down. The polynomials are the functions whose nth derivative is constantly 0. The sine and cosine functions satisfy $y'' + y = 0$ and the exponential function satisfies $y' = y$.2012-04-26
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    @Mike: or if one wishes to be a bit more inclusive, the exponential function, the sine, the cosine, and their hyperbolic counterparts all satisfy the differential equation $y^{(iv)}=y$.2012-04-26
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    How do you integrate the elementary function U(x) from edit 1?2013-03-01
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    As far as I can see, there's no reason to have a fixed, unchanging notion of "elementary function." More useful are statements like: "The least set of entities that includes the entities [...] and is closed under the operations [...], also includes the entities [...] and is also closed under the operations [...]. Furthermore, all entities in this class can be uniquely expressed in the form [...]." Substitute the word "entities" for the word "functions" and you'll get lots of interesting notions of "class of elementary functions."2013-11-04

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