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

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 . It might be argued that such description is good enough as far as mathematical names go: "the inverse of the function " seems to be clear and succint.

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

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

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.

Mon Feb 23 16:26:48 EST 1998