TPID 4

[sheldonison]

....
The answer to the last question would appear to be yes. The \( \theta \) mappings taking the tetrational to the regular superfunction of exp, and taking it to the "alternative fixed point bipolar solution" would seem to qualify. In particular, in the first case there will be singularities in the \( \theta \) mapping whenever \( \mathrm{tet}(z) \) hits a singularity of the regular Abel function. Also, between regular superfunctions, the \( \theta \) mappings taking one developed at one fixed point to another developed at another would have this property as well.

That is a wonderful observation Mike. There is an analytic real valued 1-cyclic \( \theta \) mapping going from Kneser's solution to the alternate fixed point solution. That \( \theta(z) \) is not entire, but \( \theta(z) \) is analytic if \( \Im(z)<\approx 0.38 \). This seriously complicates the effort to prove the more generalized version of the TPID#4; that any other tet_primary(z+theta(z)) is unbounded, or has singularity in the vertical strip. We need to add at least one more uniqueness criteria: that the Kneser tetration is the only one that has an slog that is analytic at the real axis, or equivalently, that tet'(z)>0 for real(z)>-2.

...The algorithm I used to generate a seed value was to start with sexp(z) from the primary fixed point, and use \( \text{sexp}(z-\sin(\frac{2\pi z} {2\pi})) \)... Then this initial approximation required an additional 42 iterations, generating ... the sexp(z) approximation around z=-1. This gave results accurate to ~32 decimal digits.

That's from the alternate fixed point post I made, #19. \( \theta(z)=\text{slog}(\text{tet}_{\text{alt}}(z))-z \) is a 1-cyclic analytic function at the real axis, even though such an alternate fixed point tet(z) doesn't have an analytic inverse at integers.... And, as you might expect, theta(z) is not entire since the fixed points are completely different in the complex plane. \( \theta(z) \) is analytic if \( |\Im(z)|\approx<0.38008 \), since tet_alt(0.38008-0.51390)~=0.3181+1.337, which is the fixed point for the Kneser primary tetration function. And at the theta(z) singularity, tet_alt(z)=tet_primary(z+theta(z)) is analytic. Should I post a graph of theta(z) near the singularity? I could also generate and post the Fourier series for \( \theta(z) \). This is a fascinating function, z+theta(z)=slog(tet_alt(z)). It circles around the integer values of tet(z) as you pass by at constant Im(z)>0. If z=-2, this means circling around a logarithmic singularity...