Solution(s) to 'power equations' I'm not sure a 'power equation' is the right name for the equation I'd like to know more about (and, specifically, about its solutions), but I don't know the 'proper' way to name it.
The simplest 'power polynomial' is : $P(x) = x^x $. The simplest 'power equation is: (1) $x^x = c$ for some $c \in \mathbb{R} $. What are the exact solutions to (1) of x (in $\mathbb{C}$) with regards to $c$, apart from the 'obvious' solutions $(x=0, c=0), (x=1, c=1)$ and $(x=2, c=4)$ and all the other solutions of the form $x^x = n^n$ for $n \in \mathbb{N}$ ?
We could extend the power polynomial: $P_{2}(x) = x^{{ax}^{bx}} + x^{cx} $. What are the exact solutions of $P_{2}(x) = d$ for $a,b,c,d \in \mathbb{R}, x\in \mathbb{C}$ ? Or perhaps I should ask what the exact form of the solutions is.
We could generalize the power polynomial even further to $P_{3}(x)$ and $P_{n}(x)$, but I don't know how to write down the latter, general polynomial.
Thanks in advance,
Max
(P.S. I References are always welcome. II If you think this question belongs to MO, please tell me).

These types of object are called super(hyper) polynomials which are polynomials of power towers or tetrations. You may want to look at http://en.wikipedia.org/wiki/Tetration for more information and references. These are fairly complicated structures and an object of current research.
