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```Archive-Name: sci-math-faq/specialnumbers/fxtofxeqx
Last-modified: December 8, 1994
Version: 6.2

Name for f(x)^(f(x)) = x

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. 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.

_________________________________________________________________

alopez-o@barrow.uwaterloo.ca
Tue Apr 04 17:26:57 EDT 1995

```