Hi.

One point that was brought up here was the idea that the regular tetration \( \mathrm{reg}_{^\infty b}[\exp_b^z](1) \) at the attracting fixed point has a natural boundary of analyticity that blocks all continuation outside the "Shell-Thron" region of bases in the complex plane.

Yet, this seems at odds with the observation that the Abel-matrix tetration works for all \( b \in (1, \infty) \) and also for some region of the complex plane around that (which I suspect, based on some crude numerical experiments, is shaped roughly like a "trumpet horn" but constricting to a point at \( b = 1 \), if you get my drift). But this would mean that it'd converge to an analytic function (note that each approximant is analytic, and a sequence of complex-analytic functions always converges pointwise to a complex-analytic function) in that region, which means that at least part of the STR boundary is not a natural boundary for this tetration. Yet it seems to agree with regular iteration in \( b \in (1, e^{1/e}] \), and so by the analytic continuation theorem it could be interpreted as an analytic continuation of regular iteration outside the STR. This makes me think:

1. the STR boundary is not a natural boundary, i.e. it has at least one gap, the end-points (on said boundary) of which are possibly delimited by the two points at which the aforementioned "trumpet curve" intersects it, or

2. Abel-matrix tetration does not actually agree with regular iteration but merely approximates it very well, so a great deal of numerical precision is needed to determine the difference (have AM and reg. been proven equivalent/nonequivalent for the appropriate range of \( b \)?).

What do you think? Certainly if (1) is true it would be very exciting, since it would mean the regular formulas already contain, in a sense, the definition of tetration for the entire complex plane -- and it would provide a good target for further research, so as to extend them out there. (2) would also be interesting, since it would raise some curious questions, like why does it come so close to regular iteration but still "miss the mark". And it would also prove regular iteration a red herring, at least insofar as making a whole-plane tetrational binary operation that all "fits together" nice and "naturally" goes.

One point that was brought up here was the idea that the regular tetration \( \mathrm{reg}_{^\infty b}[\exp_b^z](1) \) at the attracting fixed point has a natural boundary of analyticity that blocks all continuation outside the "Shell-Thron" region of bases in the complex plane.

Yet, this seems at odds with the observation that the Abel-matrix tetration works for all \( b \in (1, \infty) \) and also for some region of the complex plane around that (which I suspect, based on some crude numerical experiments, is shaped roughly like a "trumpet horn" but constricting to a point at \( b = 1 \), if you get my drift). But this would mean that it'd converge to an analytic function (note that each approximant is analytic, and a sequence of complex-analytic functions always converges pointwise to a complex-analytic function) in that region, which means that at least part of the STR boundary is not a natural boundary for this tetration. Yet it seems to agree with regular iteration in \( b \in (1, e^{1/e}] \), and so by the analytic continuation theorem it could be interpreted as an analytic continuation of regular iteration outside the STR. This makes me think:

1. the STR boundary is not a natural boundary, i.e. it has at least one gap, the end-points (on said boundary) of which are possibly delimited by the two points at which the aforementioned "trumpet curve" intersects it, or

2. Abel-matrix tetration does not actually agree with regular iteration but merely approximates it very well, so a great deal of numerical precision is needed to determine the difference (have AM and reg. been proven equivalent/nonequivalent for the appropriate range of \( b \)?).

What do you think? Certainly if (1) is true it would be very exciting, since it would mean the regular formulas already contain, in a sense, the definition of tetration for the entire complex plane -- and it would provide a good target for further research, so as to extend them out there. (2) would also be interesting, since it would raise some curious questions, like why does it come so close to regular iteration but still "miss the mark". And it would also prove regular iteration a red herring, at least insofar as making a whole-plane tetrational binary operation that all "fits together" nice and "naturally" goes.