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From: [email protected] (Alex Lopez-Ortiz)
Newsgroups: sci.math,news.answers,sci.answers
Subject: sci.math FAQ: Name for f(x)^f(x) = x
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Date: 17 Feb 2000 22:51:59 GMT
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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                                         [email protected]
http://www.cs.unb.ca/~alopez-o                       Assistant Professor
Faculty of Computer Science                  University of New Brunswick