05/23/2010, 07:54 AM
(05/23/2010, 12:05 AM)sheldonison Wrote: Has anyone generated the base e tetration from an alternative fixed point? The second fixed point (as far as I can tell) for base e is 2.0622777296+i*7.5886311785
It seems like one could develop a complex plane super function from this alternative fixed point. And from the complex plane super function, one could use one of the methods (Riemann mapping or the Cauchy method) to turn this complex super function into a real valued function with f(0)=1, and with singularities at f(-2), f(-3), etc. This would be a different tetration solution, but it would be analytic. Just curious if anyone had tried this.
- Shel
The problem with alternative fixed points is that we dont have a sickel between them. I.e. if we connect two conjugated non-primary fixed points with a straight line, then the image under \( e^z \) of this straight line overlaps itself. This is due to the imaginary part of the fixed point pair is apart more then \( 2\pi \) which makes the image of any connecting line between these fixed points, revolve around 0 at least once.
For Kneser's solution we need a region that is bounded by a line connecting the two fixed points and the image of this line.
Perhaps one (you?) could prove, that regardless how you connect a conjugated fixed point pair, that is not the primary one, the image of this line intersects itself or the connecting line; i.e. both lines never delimit a singly connected region.
