The definition of what an elementary function is is quite arbitrary (see what math.SE has to say about it) and it makes me wonder why hasn't the mathematical community added other rather natural functions to the list, for example the error function, logarithmic integral function or the Jacobi elliptic functions among many others. Is there an historical reason for this?.
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I do not know who was the first to use the term, but the rigorous DEFINITION of an elementary function is due to Joseph Liouville (Journal de l'École polytechnique, vol.14 (1833), followed by several other papers).
Roughly speaking these are functions which are obtained from algebraic functions and exponentials and logarithms by using arithmetic operations and superpositions.
Liouville's goal was to determine which integrals of elementary functions are also elementary, and which algebraic differential equations have elementary solutions. This is called sometimes Liouville's theory (or theories).
These theories had substantial development in 19 and 20-th century (Chebyshev, Hardy, Ritt, Rosenlicht, Picard, Vessiot and others)
The other functions you mention (elliptic, error function and logarithmic integral) are not elementary according to Liouville's definition. They belong to a class of "special functions", but unlike elementary functions, this class was never rigorously defined.
When some function is important enough, and its main properties have been studied in great detail, it is usually called "special", but I don't think that the class of special functions was ever seriously (mathematically) defined.
The main special functions treated in the books called "Special functions" are Gamma-function, Zeta-function, elliptic (and related) functions, and solutions of some simple linear differential equations of second order with at most 3 singular points. This set of functions was defined and studied in 19-th century. 20-th century added to the list solutions of Painleve equations and Heun equation (and its degeneracies) and sometimes various mathematicians suggest adding these functions to the list of special functions. Heun's equation has 4 singular points, and on my opinion it is early to call its solutions or solutions of Painleve equations "special functions", because our knowledge about them is still very incomplete.
So one can say that they are on the boundary of the class.
On my opinion, the class of elementary functions is interesting enough (because of the Liouville's theory) so it deserves a name. What the name exactly is, is more or less irrelevant, and everyone seems to be happy with the existing name.
Of course, one can also define elementary functions as those which are studied in high school and obtain almost the same class:-) ("Almost" because the only algebraic functions which occur in high school are those expressible in radicals).
Then this will be similar to "elementary mathematics", the term which however was never rigorously defined.
Let me also add that elementary functions have been studied in much greater detail than other special functions. For example we know how to compute (in practice!) $e^x$ or $\log x$ to billions of digits. But for most other special functions I do not now how to do this.
answered Jan 3, 2015 at 13:52
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2$\begingroup$ It's probably worth pointing out that the notion of "elementary function" used by Liouville and Chebyshev in the middle 1800s (and later by Joseph Ritt and others) includes all algebraic functions, without regard to whether they can be given in "explicit closed form". For example, the algebraic equation $y^5 - y - x = 0$ defines an algebraic function $y(x)$ (and hence an elementary function) that cannot be expressed in "explicit closed form". (This example comes from p. 55 of the 1947 9th edition of Hardy's A Course of Pure Mathematics.) $\endgroup$ Jan 3, 2015 at 16:30
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$\begingroup$ Yes, in all these definitions, elementary functions include all algebraic functions. It is a separate business, when they are represented in radicals (to which J. F. Ritt also made a significant contributiuon). $\endgroup$ Jan 4, 2015 at 8:15
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2$\begingroup$ Another tangential remark. Elementary functions in the Liouville (etc.) sense are a sub-class of analytic functions defined on a domain in the complex plane. So we exclude, for example, the function equal to 1 on the rationals and 0 on the irrationals, even though it solves the polynomial equation $y^2-y=0$. $\endgroup$ Jan 5, 2015 at 14:40