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theta and the Riemann mapping
#3
I've been playing around with Mike's pari-gp implementation of the HSB graphing system, and I thought I'd add a busy plot, showing the evolution of the superfunction (on top), to the sexp(z) function (on the bottom), with the z+theta(z) contour highlighted in yellow. I repeated the contour between the two graphs, with more details. The superfunction is exactly Periodic, with a period=4.4470 + 1.0579i; below the sexp(z) function converges to the same period as imag(z) increases. I'm using the HSB graphing system unmodified, however, I used some tricks to preserve as much accurate information as possible, when iterating exp(z) to huge numbers, while eventually treat the imaginary component as a random number, to model the chaos as the points randomly go from huge positive, to huge negative.

So this plot, in addition to showing the superfunction, and the sexp(z), also shows one unit length of the z+theta(z) mapping linking the two functions. The highlighted yellow path shows one unit length, developed by taking the inverse superfunction from z=0 to z=1. You can also see the the cyan contour from the previous iteration, going from -infinity to 0. In the HSB graphing system, negative numbers are shown in cyan, and positive numbers are shown in red, with zero being black. The larger the magnitude, the brighter the plot. For larger magnitudes approaching 1E100, the saturation increases to pure white. The rainbow in the sexp(z) function occurs at approximately sexp(2.5 to 3), as the sexp(z) function begins to swings from large positive to large negative numbers. I tried to show how z+theta(z) switches back and forth, turning around infinite number of times as necessary, as it takes an infinitely long path from the inverse_superfunction(0) to the inverse_superfunction(1). I cut off the infinite section going towards superfunction(z)=1. The yellow contour gets mapped to sexp(-1) to sexp(0). If the yellow contour is repeated, then everything above the yellow contour has a 1 to 1 bijection with the upper half of the complex plane for sexp(z). Sometime, I will post the pseudo recursive relationship, governing the pseudo repeating pattern, and the turn arounds in the contour, which occur at 1/sexp(n) for integer values of n>=5; the graph in the middle shows some of the pattern.
- Sheldon
   
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theta and the Riemann mapping - by sheldonison - 09/21/2011, 05:23 AM

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