It seems, that the eigensystem-based method works now fine.

Using the fixpoint-tracer and the values it supplies, I could construct the Bs-matrices with a few basic examples for s>e^(1/e) (or b>eta in the other notation here) based on the complex values for t and log(t).

The constructed matrices matched perfectly the naive-versions for integer-tetration Bs and Bs^h.

This settles then the continuous tetration for all s>e^-e (except the known singularities by the method)

The complex fixpoints t=h(s) for s>e^(1/e) are taken from the branch, which reaches into the real halfplane with real(t)>1 (the other branches there have real(t)<1) and is shown in the graphs at Fixpoint principal branch.

Phew. ;-)

It seeems, what remains is now consideration of numerical aspects and optimization of summing, where intermediate non-converging series occur.

Gottfried

Using the fixpoint-tracer and the values it supplies, I could construct the Bs-matrices with a few basic examples for s>e^(1/e) (or b>eta in the other notation here) based on the complex values for t and log(t).

The constructed matrices matched perfectly the naive-versions for integer-tetration Bs and Bs^h.

This settles then the continuous tetration for all s>e^-e (except the known singularities by the method)

The complex fixpoints t=h(s) for s>e^(1/e) are taken from the branch, which reaches into the real halfplane with real(t)>1 (the other branches there have real(t)<1) and is shown in the graphs at Fixpoint principal branch.

Phew. ;-)

It seeems, what remains is now consideration of numerical aspects and optimization of summing, where intermediate non-converging series occur.

Gottfried

Gottfried Helms, Kassel