complex base tetration program

[mike3]

Neat stuff. I'm curious: how did you get the method to work? That is, when we take the Abel function of the Taylor approximation at z = 0 for the Fourier integral to get the warping map (the \( \theta(z) \) mapping), how do you make sure that the values going into the Abel function are within its range of convergence? That seems to be the trick part that makes it difficult to extend the method to various other complex bases.

Hey Mike,

Thanks for commenting. As far as the Abel function, goes, I do that evaluation over a unit length from -0.5 to 0.5, and I extended the imaginary delta to 0.175i, for the upper superfunction theta, and -0.175i for the lower superfunction eta. This helps remove ambiguity on "which logarithm branch" to use, and I also have some code tweaks to make sure the inverse Superfunction (or Abel function) is mapping to the same period for all sample points. The other problem is deciding how accurate the Schroder function needs to be. The program iterates logarithms (for the repelling case), before evaluating the Schroder function, until the sample point is within some bounds of L, where the Schroder function is accurate. One difficulty is how to know what to use as the bounds, and I'm still tweaking that, with adjustments for bases near eta, and will put an update of the code online shortly. Finally, I initialize and renormalize the superfunction so that B^sexp(-0.5+/-0.175i)=sexp(0.5+/-0.175i), which at least gives continuity to the theta calculations. This dramatically improves the numer of bits precision improvement I get at each iteration.

What do you mean by "mapping to the same period"?

Right now, I'm numerically investigate the branchpoint singularity at eta, which is incredibly mild, as you and others have noticed, and I will post some surprising results about that.
- Sheldon

What did you find?