Search the FAQ Archives

3 - A - B - C - D - E - F - G - H - I - J - K - L - M
N - O - P - Q - R - S - T - U - V - W - X - Y - Z
faqs.org - Internet FAQ Archives

sci.math FAQ: f(x)^f(x)=x


[ Usenet FAQs | Web FAQs | Documents | RFC Index | Sex offenders ]

See reader questions & answers on this topic! - Help others by sharing your knowledge
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


User Contributions:

Comment about this article, ask questions, or add new information about this topic:

CAPTCHA


[ Usenet FAQs | Web FAQs | Documents | RFC Index ]

Send corrections/additions to the FAQ Maintainer:
alopez-o@neumann.uwaterloo.ca





Last Update March 27 2014 @ 02:12 PM