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).

Maybe you should read about the Lambert W-function which gives the solutions to expressions like $z=x^x$. However, I am not sure what to do in case of such a "power tower" like $P_2$.
