andydude Wrote:...I just found another point (probably found before) on the outer branches, one that has many advantages over the point of intersection. This point is \( (x=\frac{1}{2}, b=\frac{1}{16}) \) which is 2-periodic in the sense:as can be seen in this plot:
- \( (1/16)^{(1/2)} = (1/4) \)
- \( (1/16)^{(1/4)} = (1/2) \)
[attachment=198]
Andrew Robbins
I found a similar result with "my" approximation, including your two 2-peridic points, obviously referred to a different base. Interesting!
@ IVARS
Please see the attachment, in which I try to imagine the situation represented in a 3-dimensional environment (y,b,x). The plots of y=b#x are the (appromimated and simulated) real projectionns on the yx plane of the y=d#x complex helicoidal "functions". The transition (yellow, off-limit) zone on the yb diagram is (in my ... simple opinion) the graph of the max-min span of the undetermined y values, when x -> oo. Persintent oscillations. Difficult to describe, ... see the plots.
GFR