Solving for *f* one finds a ``continued fraction"-like answer

*f(x) = (log x)/(log(log x)/(log(log x)/(log ...)))*

This question has been repeated here from time to time over the years, and no one seems to have heard of any published work on it, nor a published name for it.

This function is the inverse of *f(x) = x^x*. It might be argued that
such description is good enough as far as mathematical names go:
"the inverse of the function *f(x) = x^x*" seems to be clear and succint.

Another possible name is *lx(x)*. This comes from the fact that
the inverse of *e^x* is *ln(x)* thus the inverse of *x^x* could be
named *lx(x)*.

It's not an analytic function.

The ``continued fraction" form for its numeric solution is highly
unstable in the region of its minimum at *1/e* (because the graph is
quite flat there yet logarithmic approximation oscillates wildly),
although it converges fairly quickly elsewhere. To compute its value
near *1/e*, use the bisection method which gives good results. Bisection
in other regions converges much more slowly than the logarithmic
continued fraction form, so a hybrid of the two seems suitable.
Note that it's dual valued for the reals (and many valued complex
for negative reals).

A similar function is a built-in function in MAPLE called *W(x)*
or Lambert's *W* function.
MAPLE considers a solution in terms of *W(x)* as a closed form (like
the erf function). W is defined as *W(x)e^(W(x)) = x*.

Notice that *f(x) = exp(W(log(x)))* is the solution to *f(x)^f(x) = x*

An extensive treatise on the known facts of Lambert's W function is available for anonymous ftp at dragon.uwaterloo.ca at /cs-archive/CS-93-03/W.ps.Z.

Fri Feb 20 21:45:30 EST 1998