See reader questions & answers on this topic!  Help others by sharing your knowledge ArchiveName: scimathfaq/specialnumbers/fxtofxeqx Lastmodified: 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 builtin 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 /csarchive/CS9303/W.ps.Z. _________________________________________________________________ alopezo@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:
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