tetration from alternative fixed point
(06/28/2010, 11:03 PM)sheldonison Wrote: ....To generate these contours lines in the SuperFunction, I used the inverse super function with img(z)=-3pi*i contour with real(z) varying from . Then I graph the exponent of that contour line, which is repeated one unit away, with img(z)=0 and real(z) varying from - to 0, and the exponent of that contour line with img(z)=0 and real(z) varying from 0..1. The three contours are disconnected from each other, which I think you were pointing out to begin with. The Reimann mapping will not be analytic at the boundary points.
- Sheldon
[Image: sexp_fixed_2.gif]
The complex periodicity is incorrect in this graph, which effects the "alternative 3*pi*i contour" graph. For the period I incorrectly used:
period = 2Pi*i/L
The correct value is
period = 2Pi*i/(L-2*Pi*I)

This is because the correct equation for the periodiicty is 2*Pi*i/ln(L)
Because L>2Pi*I, the primary ln(L)=L-2*Pi*I. The correct complex period=1.3769+2.1751*I.

I haven't verified the rest of the graph, but otherwise, I'm still using the same equations as I used when I made this graph, and those equations should have given correct values for the other complex contours. By the way, the equations are posted here. I found the problem when I wrote a pari-gp script for the secondary fixed point. With the fix, the alternative contour no longer fits snugly against the primary i=0 contour. I'm still interested in seeing if there's any way to Riemann map the contours back to a well defined real axis, but I still assume that it is not possible.
- Sheldon

Messages In This Thread
RE: tetration from alternative fixed point - by sheldonison - 08/13/2010, 03:53 PM

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