$x^x = y$, express $x$ in terms of $y$

Question

$x^x = y$

Express $x$ in terms of $y$.

I'm particularly interested in the solution where $x$ is real, but complex would be interesting too.

Any ideas? Thanks.

$x$ in terms of $y$ is not a valid function. It would be a correspondence. — Erik M, Mar 21 '14 at 21:20
Thank you, Erik.
Is there a trivial way of showing it is not a valid function?

(Also, is it not a function in the sense that there may be multiple values of $x$ for certain values of $y$, or in some other sense?) — Mysterious Entity, Mar 21 '14 at 21:24
I'm surprised it is not a valid function. The graph in Wolfram Alpha looks like it could be.
link — Mysterious Entity, Mar 21 '14 at 21:28
Yes, that's precisely the reason (there exists a $y$ that more than one $x$ value can generate). Odd, it seems Wolfram Alpha is sensitive to the order in which you enter the equation, see http://www.wolframalpha.com/input/?i=y%20=%20x%5Ex — Erik M, Mar 21 '14 at 21:33
Whenever you see a $x^x$, think of Lambert's $\mathrm W$ function. — Hakim, Mar 22 '14 at 16:33

score 6 · Answer 1 · answered Mar 21 '14 at 21:29

6

Use Lambert's W function:

In mathematics, the Lambert W function, also called the omega function or product logarithm, is a set of functions, namely the branches of the inverse relation of the function $z = f(W) = We^W$ where $e^W$ is the exponential function and $W$ is any complex number. In other words, the defining equation for $W(z)$ is

$$z = W(z)e^{W(z)}$$

for any complex number $z$.

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answered Mar 21 '14 at 21:29

draks ...

18,449

Thank you! Sorry I don't have enough reputation to vote you up. I hope others will. – Mysterious Entity Mar 21 '14 at 22:03
@MysteriousEntity You're welcome and welcome to Math.StackExchange as well.. – draks ... Mar 21 '14 at 22:07

$x^x = y$, express $x$ in terms of $y$

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