First, these include polynomials, i.e. functions which are compositions of addition, z^n and multiplication by complex constants. We can say that they are solutions of the following linear ODE: y^{(N)}=0, i.e. N-th derivative of y(z) is zero. Next we consider quotients R(z)=P(z)/Q(z) where P(z) and Q(z) are polynomials these are called rational functions.

Secondly, we include trigonometric functions: Cos(z), Sin(z). We have several alternative analytical descriptions of them. We can define it as power series or as solutions of y" + y = 0, with corresponding initial problem: for Cos(z) it is y(0)=1,y'(0)=0 and for Sin(z) it is y(0)=0, y'(0)=1. Then we can define Tan(z), and Cotan(z) as usual using Sin(z) and Cos(z).

Third function is exponential function Exp(z) which again can be defined either by the power series or as the solution of the following initial problem: y'=y, y(0)=1.

Note that if we allow complex variable and consider solution of y' + ay = 0 where a is complex number then the solution is y=Exp(-az), and using Euler formulas we can define Sin(z) and Cos(z) using Exp(z).

Finally we consider inverse functions to z^n, Cos(z), Sin(z), Exp(z), Tan(z), Cotan(z). The most important is Log(z) - the inverse to Exp(z).

Then we begin to make all compositions of previously defined functions and repeat procedure.

What can we say about this class of functions?

First, they are analytic on its domain of definition. Second, this class is closed with respect to the differentiation: if f(z) "elementary" then f'(z) is "elementary".

Third, this class IS NOT closed under integration. The simplest examples are Sin(z^2), Sin(z)/z, Exp(z)/z, Exp(-z^2). These are well-known but I catch myself that I don't know how to prove this! It hasn't been taught to me and I haven't met this in textbooks! It looks like this is not popular subject and people don't care about this!

What else I know. Nothing is coming in mind! Pretty surprisingly!

Yes! I recall that polynomials are dense in the space of continuous functions and even in smooth metrics. And polynomials is relatively "small" subclass. Thus elementary functions might serve as good approximations.

But how good they are? Anybody interested in this?

Another interesting question - as we discussed at the beginning each elementary function can be described as the unique solution of some initial value problem for some Algebraic ODE. Thus any composition has to be also a solution of some Algebraic ODE(possibly non-linear). Can we describe this class of ODE's explicitly?

Anyway in my opinion it looks that we know too little about "elementary" functions. We need to make it up!