To make things definite, let's narrow them and call transcendental equation of the form

$$f(x) = 0$$

where $f$ is a real elementary function in the usual sense. For example

[$$\cos(\pi x) + x^2 = 0$$](https://math.stackexchange.com/questions/146131/algebraic-solution-to-cos-pi-x-x2-0)
or
$$a = x \tan x$$

Is there a general way to solve such equations except for numerics? That is to produce an expression$$x\_0 = \text{RHS}$$
where $\text{RHS}$ does not depend on $x\_0$ and can be actually computed? Maybe in a form of series or something similar, not in "finite terms".

There is of course a question about the existance of solutions, it would be nice if the form of $\text{RHS}$ incorporatted the answer to it.

In the formulation above the problem seems to be equivalent to finding local inverse function.