complex base tetration program

[sheldonison]

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.....

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

What I was refering to was the last step in evaluating the inverse superfunction (or Abel function) to generate theta, after evaluating the Schroder function, and adjusting for iterated logarithms. This last step is:
\( \log(\text{Schroder}(z-L))/\log(L\times\log(\text{base})) \)
This logairthm of the Schroder function can be ambiguous, especially if fixup is required due to the accuracy range of the Schroder function. And the easiest fix is to compare adjacent points for the inverse superfunction, and to add or subtract the period so that the adjacent points are near each other.

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?

I'm still investigating and learning, and will post more details later. The experiment, is to develop taylor series for all of the coefficients of sexp_b(z), developed around complex sexp(z) in the neighborhood of base=2. The conundrum, which I'm beginning to be able to explain, is that the numerical results are way too good, and its hard to see the effects of the branch point at eta. In fact, numerically convergence seems limited by the more distant singularity for sexp_b(z), at b=1, and not at all by the closer singularity at eta. So far, I have a taylor series for all the coefficients for the neighborhood of base=2, and the resulting sexp(z) is accurate to 33 decimal digits. At eta, it is accurate to 31 decimal digits. Theory says the extrapolated sexp_b(z) function shouldn't even converge if the radius is outside 2-eta, but for the truncated series, that is not at all the case. For example, results for base e are still accurate to 19 decimal digits, and for base 1.3, the results are accurate to about 20 decimal digits. Precision declines nearly linearly from the sample radius (2-eta)=~0.555, towards the more distant singularity at b=1, with radius=1. More details later. I also put a new version of the code in the first post, with improvements near eta, and convergence for ultra high precision results for >p 67.
- Sheldon