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This question has been bugging me since high school where I was told "not to be concerned with such matters", but years later I still haven't found a satisfying answer.

The question is really simple:

$ x^x = y $ given $y$, where $y \in \mathbb{R}$ solve for $x$ (analytically)

JoelKuiper
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    This question also bothered me in a high school before I've installed Maple. I'd rather suggest you ask yourself another question - what is an analytical solution? As soon as we can show that $f(x) = y$ has a unique solution for all $y$, we can define $x = g(y)$ unambiguously where $g$ is by definition the solution of the equation $f(x) = y$. At the same time, it may tell a little to you how to compute $g$. If you want to express $g$ using a finite composition of elementary functions, I guess it is not possible. – SBF Dec 20 '13 at 13:10
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    (ctd.) but at the same time, who said that the function $\cos(\sin^{\sin x})\cdot\mathrm e^{1/(x^2+x)}$ is easier to deal with? I also guess, this question feets better MSE and is likely to be moved there soon. – SBF Dec 20 '13 at 13:12
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    No solution exists in terms of elementary functions, but in some sense it can be solved by: http://en.wikipedia.org/wiki/Lambert_W_function . Check out example 2 (and 3). – Daniel Soltész Dec 20 '13 at 13:13
  • To me, and I can be wrong, if the function W(y) exists, then the analytical solution exists. – Claude Leibovici Dec 20 '13 at 14:09
  • So my follow up question is, why is there no elementary function or as @ClaudeLeibovici and Daniel said how would you solve the W(y) function? – JoelKuiper Dec 20 '13 at 21:44
  • Because the inverses of elementary functions are known. In this case, we are dealing with a function that is neither polynomial (constant exponent), nor exponential (constant base), let alone trigonometric (exponential of imaginary argument), etc. – Lucian Dec 20 '13 at 22:11
  • I can solve this for $y=e^{-\pi/2}$ :) – Agol Dec 21 '13 at 04:50
  • @JoelKuiper. As you probably saw, the solution of x^x = y is x = Log(y) / W[Log(y)]. Now you take us in a highly philosophical debate. What is the limit of elementary functions ? Thanks for this question. Cheers. – Claude Leibovici Dec 21 '13 at 08:14

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