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If $y^y=x$, can y be expressed as a function of x? Specifically, I am finding the solution to a PDE where the most general solution is $u=t^{-\frac{1}{2}} f(x,t)$ and $$\LARGE f^f=Ce^{\frac{-x}{2\sqrt{t}}} $$ Any help will be appreciated!

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marked as duplicate by Thomas Andrews, Hans Lundmark, T. Bongers, Live Forever, Lucian Jul 20 '14 at 22:59

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Yes, the Lambert W function will do the job. There are some old posts on this but I can't seem to find them right now – TylerHG Jul 20 '14 at 21:36
up vote 5 down vote accepted

If $x=y^y$, take $\log$ of both sides to get: $y\log y=(\log y)e^{\log y} = \log x$ and thus $\log y = W(\log x)$ and thus $y=e^{W(\log x)}$, where $W(z)$ is the Lambert-W function satisfying $W(z)e^{W(z)}=z$.

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