There are two definitions for *elementary functions*. The first one appears to be *elementary*, but leads to all sorts of "guesses". The second one is not *elementary* in that it involves some "non-elementary concepts" in abstract algebra.
First one is a *practical one*. A function of a finite number of real (or complex) variaible is called *elementary* if it is the *composite* built up from a finite number of the following types of functions:
*algebraic*, exponential, logarithmic, trignometric, or inverse trigonometric.
Roughly, they comprise the most common types of function in elementary calculus. The difficulty of this definition is that elementary functions need not be single-valued, nor everywhere defined. There is also something that had not been said: where is the meaning of *composite*. For example: is sin x + arc tan y + sqrt( z^3 + ln(u)/exp(v/w)) an "elementary function" or not.
Second one is more systematic and due to J. Liouville (1838). He defined elementary function by assigning a *class* to each of them.
Class 0 comprised of all *algebraic functions* of a finite number of complex variables.
This can be concretely defined via the concept of a *field* in algebra. Namely, begin with the complex numbers C and form the field C(x_1, ..., x_n, ....) of rational functions in a countably infinite set of variables x_i. Use the general theory of fields to construct the algebraic closured C_0 of this field. Its elements are the algebraic functions of class 0.
To go to elementary functions of class 1, we introduce functions exp(z) and ln(z).
Now, inductively, we assume that elementary functions of class at most n-1 has been defined (n a positive integer), let g(t), g_j(w_1,...,w_n) be a finite collection of elementary functions of class at most 1, j = 1, ..., m, and let f(z_1,...,z_m) [same m] be an elementary function of class at most n-1. Then the composite functions: g(f(z_1,...,z_m) and f(g_1(w_1,...,w_n),...,g_m(w_1,...,w_n) (and only such are called elementary functions of class at most n.
While it is true that the derivative of an elementary function is again an elementary function, it is not true that the integral of an elementary function is necessarily elementary. Similarly, the "inverse function" of an elementary function need not be elementary. The entire subject is now a part of "differential field theory"--a part of algebra. Liouville carried out a deep and detailed analysis where the integral of an elementary function is again an elementary function. What happened prior to Liouville was that mathematicians spoke of "non-elementary functions" without any "agreement" and controversies abound. The main point was that the integrals of *elementary functions* are very interesting. This is the *inverse* process to *differentation*. Many physical problems began with elementary functions but ends up asking about the knowledge of the integral of these elementary functions.
In the rush to de-emphasize the techniques of integration on the grounds that integrals can be evaluated using tools like the graphing calculators, and at the same time of trying to emphasize the *concept*, over *blind calculations*, one runs into the paradox: we are throwing away the basis of many of the important concepts we are trying to teach.
Han Sah, email@example.com
P.S. While Euler's formula and the Konigsburg Bridge Problem were often viewed as the fountain heads of topology, one other fountain head of topology arose from the geometric view to the algebraic functions --in particular, algebraic functions of one variable. The classic in this area was written by H. Weyl (the author of Symmetry) around 1927--the idea of a Riemann surface, English translation of the German original.