theta and the Riemann mapping

[sheldonison]

The equation to convert the theta(z) unit disk to Kneser's Riemann mapping unit disk is as follows, where u(z) is \( \theta \) wrapped around the unit disk.
....
\( \text{RiemanMapping}(y) = y \times \exp(u(y)\times 2\pi i) \)

This can also be reversed, to derive the unit circle theta(z) function from the Riemann mapping.
\( u(y) = \frac{\log (\text{RiemanMapping}(y)/y)}{2\pi i} \)
....
Both the unit circle version of theta(z) and the Riemann mapping unit circle are defined everywhere in the unit circle, except for the singularity at z=1.

Using these equations, I have succeeded in recreating all of the graphs in Jay's post, as well as some new ones of my own which I will post later.
- Sheldon Levenstein

From my point of view, the biggest obstacle to using the theta(z) in my own program to generate numerical results for sexp(z) is the really nasty singularity that theta(z) has at integer values of z. This is probably also the biggest obstacle to using the Riemann mapping in Kneser's construction to generate numerical results for sexp(z). Both have fascinating very complex singularities, as you superexponentially approach integer values for sexp(z).

This is the contour for theta(z)+z, going through 5 iterations, from z=-2 to z=3 \( \text{superf}^{-1}(\text{sexp}(z)) \), where we approach within sexp(3.5) or 1.6E-78 of the integer values of z.
   

This is the corresponding Riemann unit circle mapping, approaching within 1.6E-78 of the singularity. Surprisingly, At this level, there is still a visible discontinuity in the Riemann mapping unit circle! Later in this post, I post the numerical values for 100 terms of the Taylor series of the Kneser Riemann mapping unit circle function.
   

Going back to the theta(z)+z contour, now I fill in the detail, getting superexponentially closer and closer to the singularity at sexp(z=-1)=0. The earlier plot above approached to within sexp(z)=1.6E-78. In green, the first extension goes 1/sexp(3.5) to 1/sexp(4.5). For comparison, a googolplex is roughly sexp(4.53). But the detail increases as we superexponentially approach zero. The next extension, in red goes from 1/sexp(4.5) to 1/sexp(5.5). Sexp(5.5) is a number too larger to meaningfully describe. The next extension, in green, goes from 1/sexp(5.5) to 1/sexp(6.5). And the next extension, in red goes from 1/sexp(6.5) to 1/sexp(7.5). And finally, in green 1/sexp(7.5) to 1/sexp(8.5). I used superexponential approximations to calculate these plots, which was actually quite a difficult calculation! Underneath, is the exact corresponding theta(z) plot. Notice that theta(z) function will slowly approach +real infinity, and -imag infinity, as the function superexponentially approaches the singularity at zero. Also, notice that the singularity becomes ever more and more complicated, winding and unwinding along repeated paths superexponentially close to paths that it has already followed.
   

For refeence, this is the chi-star contour plot, through 5 iterations, from z=-2 to z=3 \( \text{Schroeder}(\text{sexp}(z)) \), where we approach within sexp(3.5) or 1.6E-78 of the integer values of z.
   

I'm not an expert on Riemann mappings, and I've never used any of the kernals that can be used to calculate a Riemann mapping. My own approach is to avoid the singularity, where the functions are more well behaved. For my kneser.gp program, I iteratively calculated the theta(z) mapping at imag(z)=0.12i, instead of at the real axis. This is far enough away from the singularity to avoid all of the numerical difficulties. So, I thought to myself, why not use the exact same idea to iteratively calculate Kneser's Riemann mapping? So, I modified a version of my "kneser.gp" program, to use the Riemann mapping in place of the unit circle theta function, using the equations from the first post in this thread, for sexp(z) base "e". I used the code I developed in kneser.gp, to iteratively generated the Riemann mapping approximations and sexp(z) approximations from each other, using the Riemann mapping unit circle function to give increasingly more accurate approximations for sexp(z) for imag(z)>=0.12i. After fifteen iterations, the result, was a 110 term taylor series for the Riemann mapping unit circle, accurate to 32 decimal digits, for imag(z)>0.12i, along with a taylor series for sexp(z), centered at z=0, and also accurate to 32 decimal digits. This compares with thirteen iterations required for the theta(z) mapping, so this Riemann mapping approach appears to be a little bit less efficient than the theta(z) approach. Here is the resulting Riemann mapping unit circle taylor series that I iteratively generated. The absolute value of the Taylor series terms is slowly decreasing until the 98th term, so the terms beyond that have no numerical value, and are doubling due to random noise in the numerical calculations, since 0.12i corresponds to a radius of 0.47 for the unit circle function.

Now that I know definitively that theta(z) is a completely different function than the Riemann mapping unit circle, I will update the naming conventions used in the next version of my kneser.gp program update, to eliminate any references to the Riemann mapping unit circle. I realize that anyone familiar with Kneser's approach must have been confused by my naming conventions.
- Sheldon Levenstein

Code:

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