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### Subject: sci.math FAQ: Name for f(x)^f(x) = x

This article was archived around: 17 Feb 2000 22:51:59 GMT

Archive-name: sci-math-faq/fxtofx
Last-modified: February 20, 1998
Version: 7.5

Name for f(x)^(f(x)) = x
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.
_________________________________________________________________
--
Alex Lopez-Ortiz alopez-o@unb.ca
http://www.cs.unb.ca/~alopez-o Assistant Professor
Faculty of Computer Science University of New Brunswick