Hi.

I think I may have discovered an alternate solution, not the regular iteration(!), for the tetration of the bases in the "convergent" or "Shell-Thron Region" (STR) of the complex plane, that resembles the "holomorphic in the right halfplane and decays to two fixed points at imag infinity" solution for bases outside that region. After all, this would make sense, no -- there doesn't seem to be any reason there should be a "bump!" when we hit the STR boundary that causes us to lose that behavior, after all. Indeed, I'd expect the only "bumps" to be at the points 0, 1, and perhaps , which would be singularities or branch points.

Consider the complex base , of which such a solution was constructed via the continuum sum method with a Fourier approximation (have had no luck using the Cauchy integral at complex bases). See here:

http://math.eretrandre.org/tetrationforu...hp?tid=437

The graph at the complex plane looks like:

(scale is +/-10 on both axes)

The limits at are and (where the +/- indicates positive/negative imaginary infinity, respectively). Using Newton's method to approximate fixed points of exponential and gradually stepping the base backward parallel to the real axis (using the last fixpoint as a guess to find the one for the "stepped-back" base), we get that for , which is in the STR, that this solution should approach and .

And the most astonishing thing is that this solution actually appears to exist! An experiment with the same continuum-sum Fourier method, taking the assumption the solution is holomorphic in the right half-plane (implementing the Fourier expansion along the imaginary axis is equivalent to this and must exclude the regular as it has singularities that are at minimum arbitrarily close to the axis, which'd mean a Fourier expansion of zero convergence width), and then gradually stepping back the base, using the already-calculated solution as an initial guess, we get this:

This certainly seems to make more sense as opposed to the "regular" solution, which looks like this:

Note how it lacks the fractal zones and is not holomorphic in the right half-plane (note the cut lines and singularities) -- features the tetrational outside the STR had.

I'd hypothesize that this "alternative" solution is what the analytic continuation of the "Cauchy"/"Kneser" tetrational in the base to the STR would look like. Also, notice the strange branchpoints and cutlines in the left halfplane. The cuts and points are probably too complicated to give a simple description. If tetration ever is developed to the point where it could be a "real" special function, it'd probably be the most exotic.

I believe this solution of tetration will not be real at , and will be a branch point, along with and (suggest cut at .). However, due to the fractal nature, it'll likely satisfy for all but the singularities. Oddly enough, I can't seem to get the construction via this sum method to work for bases nearing the negative real axis for some reason or too far in to the left half-plane -- not sure why. But I don't see why a function which has similar imaginary-asymptotic properties can't exist there.

Regular was appealing, since it gave real-at-the-real-axis solutions for , but appears to have a natural boundary at the STR boundary, and so is worthless for creating a full-scale tetrational on the overall complex plane.

Puzzling observations: why does Carleman-power seem to give regular at and Cauchy-like when ??? And why does continuum sum seem to yield regular, too, when is on the real axis in that range? What is going on here? What is this strange "affinity" these two seem to have for each other?

I think I may have discovered an alternate solution, not the regular iteration(!), for the tetration of the bases in the "convergent" or "Shell-Thron Region" (STR) of the complex plane, that resembles the "holomorphic in the right halfplane and decays to two fixed points at imag infinity" solution for bases outside that region. After all, this would make sense, no -- there doesn't seem to be any reason there should be a "bump!" when we hit the STR boundary that causes us to lose that behavior, after all. Indeed, I'd expect the only "bumps" to be at the points 0, 1, and perhaps , which would be singularities or branch points.

Consider the complex base , of which such a solution was constructed via the continuum sum method with a Fourier approximation (have had no luck using the Cauchy integral at complex bases). See here:

http://math.eretrandre.org/tetrationforu...hp?tid=437

The graph at the complex plane looks like:

(scale is +/-10 on both axes)

The limits at are and (where the +/- indicates positive/negative imaginary infinity, respectively). Using Newton's method to approximate fixed points of exponential and gradually stepping the base backward parallel to the real axis (using the last fixpoint as a guess to find the one for the "stepped-back" base), we get that for , which is in the STR, that this solution should approach and .

And the most astonishing thing is that this solution actually appears to exist! An experiment with the same continuum-sum Fourier method, taking the assumption the solution is holomorphic in the right half-plane (implementing the Fourier expansion along the imaginary axis is equivalent to this and must exclude the regular as it has singularities that are at minimum arbitrarily close to the axis, which'd mean a Fourier expansion of zero convergence width), and then gradually stepping back the base, using the already-calculated solution as an initial guess, we get this:

This certainly seems to make more sense as opposed to the "regular" solution, which looks like this:

Note how it lacks the fractal zones and is not holomorphic in the right half-plane (note the cut lines and singularities) -- features the tetrational outside the STR had.

I'd hypothesize that this "alternative" solution is what the analytic continuation of the "Cauchy"/"Kneser" tetrational in the base to the STR would look like. Also, notice the strange branchpoints and cutlines in the left halfplane. The cuts and points are probably too complicated to give a simple description. If tetration ever is developed to the point where it could be a "real" special function, it'd probably be the most exotic.

I believe this solution of tetration will not be real at , and will be a branch point, along with and (suggest cut at .). However, due to the fractal nature, it'll likely satisfy for all but the singularities. Oddly enough, I can't seem to get the construction via this sum method to work for bases nearing the negative real axis for some reason or too far in to the left half-plane -- not sure why. But I don't see why a function which has similar imaginary-asymptotic properties can't exist there.

Regular was appealing, since it gave real-at-the-real-axis solutions for , but appears to have a natural boundary at the STR boundary, and so is worthless for creating a full-scale tetrational on the overall complex plane.

Puzzling observations: why does Carleman-power seem to give regular at and Cauchy-like when ??? And why does continuum sum seem to yield regular, too, when is on the real axis in that range? What is going on here? What is this strange "affinity" these two seem to have for each other?