Also, fidding around with this some more, it seems that we cannot continue below \( e^{-e} \) in one step, suggesting that it is a singularity (likely a branchpoint).Trying to continue into the pseudocircle with this method from some direction in the complex plane also does not seem to work well, at least with only one step, suggesting lots of singularities/branchpoints in this area. This is Weird!!! Yet we can go up the real axis past \( e^{1/e} \)...
It seems possible that certain Hopfian bifurcation points (of the integer tetrations) are also branchpoints, but not all (note that if one looks at the tetration fractal, the point \( b = e^{1/e} \) is an accumulation point of Hopfian points, yet if they were all singularities it would be impossible to analytically continue to \( b \g e^{1/e} \), yet this seems to work OK.). A graph might be useful in figuring out what's going on here, but solving the regular iteration near the edges of the Shell-Thron region is difficult, esp. if we want a high-resolution plot. Perhaps if there was some way to evaluate the regular iteration on the edge of the region? But this is tough -- see the thread about regularly iterating \( b = e^{-e} \)... Note we can parameterize the edge using exponentials, however (see here: http://math.eretrandre.org/tetrationforu...11#pid3011), so obtaining points on it is not a problem.
All these wild things make me think that if tetration is ever developed to the point where it could be a "real" special function like the gamma function, error function, zeta function, hypergeometric, Lambert W, etc. it'd be one of the most exotic yet. Especially considering the "fractal" structure (looks like the dynamical Julia set of the exp. function!) it produces for bases like those greater than \( e^{1/e} \) when it is plotted for complex values of the tower.
It seems possible that certain Hopfian bifurcation points (of the integer tetrations) are also branchpoints, but not all (note that if one looks at the tetration fractal, the point \( b = e^{1/e} \) is an accumulation point of Hopfian points, yet if they were all singularities it would be impossible to analytically continue to \( b \g e^{1/e} \), yet this seems to work OK.). A graph might be useful in figuring out what's going on here, but solving the regular iteration near the edges of the Shell-Thron region is difficult, esp. if we want a high-resolution plot. Perhaps if there was some way to evaluate the regular iteration on the edge of the region? But this is tough -- see the thread about regularly iterating \( b = e^{-e} \)... Note we can parameterize the edge using exponentials, however (see here: http://math.eretrandre.org/tetrationforu...11#pid3011), so obtaining points on it is not a problem.
All these wild things make me think that if tetration is ever developed to the point where it could be a "real" special function like the gamma function, error function, zeta function, hypergeometric, Lambert W, etc. it'd be one of the most exotic yet. Especially considering the "fractal" structure (looks like the dynamical Julia set of the exp. function!) it produces for bases like those greater than \( e^{1/e} \) when it is plotted for complex values of the tower.
