Steve W. writes:
<< This is the tower-of-exponentials I'm playing with now:
(*) fe(x) = 2 sinh( fe( x - tanh( x/2 ) ) )
As x --> +oo, fe(x) --> e^f( x-1 ),
should be --> e^fe( x-1 ),
as x --> -oo, fe(x) --> -e^-f( x+1 ) (or just -f(-x)), and
should be --> -e^-fe( x+1 ) (or just -fe(-x)), and
as x --> 0, fe(x) --> 2 fe( x/2 ).
To calculate despite the circular definition, I assume a straight line with some slope when abs(x) < 2^-27, where sinh and tanh are linear enough that the location of the seam is invisible to floating point.
From: Dan Asimov <dasimov@earthlink.net>
Is it clear that the functional equation I've labeled (*) has a unique solution fe ?
It's not unique. Lots of crazy solutions, but maybe this is unique: First pick an arbitrary initial slope, then take The limit of fe( x, init_slope ) = x * init_slope if |x| <= cutoff 2 sinh( fe( x - tanh( x/2 ) ) ) if |x| > cutoff as cutoff --> 0 (from above). I'm not even sure that that converges. The Python function is piecewise as above. There's a discontinuity (not just discontinuous derivatives) at the ends of the line segment, and at every echo of the ends, but the first one is on the order of one part in 2^-54, under the floating point noise, and the echoes of it are smaller than the echoes of that noise. --Steve