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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,


(P.S. I References are always welcome. II If you think this question belongs to MO, please tell me).

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Lambert's W is a start... but beyond there I'm stumped... – Tom Boardman Aug 10 '10 at 13:23
@ Tom and MRA: thanks, this a good start indeed. – Max Muller Aug 10 '10 at 13:26
There is no reason to expect that such equations have solutions in terms of familiar functions. – Qiaochu Yuan Aug 10 '10 at 16:45
There is also no reason to expect that anyone has ever studied this problem. These kinds of equations don't, to my knowledge, appear naturally in any problem. – Qiaochu Yuan Aug 10 '10 at 17:08
x=0, c=0 isn't an "obvious solution." The expression $0^0$ is indeterminate. If anything, you should use $0^0=1$ here since $\lim_{x\rightarrow 0} x^x = 1$. – Corey Jun 27 '11 at 2:32

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 $P_2$.

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

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In regards to the first solution problem, $x^x=c$, you can isolate the $x$ using the Lambert W function as follows:$$x^x=c$$$$x=c^{1/x}$$$$1=\frac1xc^{\frac1x}=\frac1xe^{\ln(c)\frac1x}$$$$\ln(c)=\ln(c)\frac1xe^{\ln(c)\frac1x}$$$$W(\ln(c))=\ln(c)\frac1x$$$$x=\frac{\ln(c)}{W(\ln(c)}=e^{W(\ln(c))}$$

For higher power towers, we require special conditions or situations for a solution to exist in a sort of closed form using the Lambert W function.

Which means that if we can't isolate $x$ in a simple higher power tower like $$x^{x^x}=c$$Then we probably can't do it for your situation.

I will also note that a solution to $$x^{x^{x^{\cdots x}}}=c$$$$x=\sqrt[c]{c}$$

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