08/13/2010, 03:53 PM
(This post was last modified: 08/13/2010, 04:01 PM by sheldonison.)

(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.The complex periodicity is incorrect in this graph, which effects the "alternative 3*pi*i contour" graph. For the period I incorrectly used:

....

- Sheldon

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