11/12/2009, 07:08 PM
Interesting, that it is the natural boundary (it would remind me of the situation with the Jacobi theta functions in the nome \( q \): the (unit, I think) circle is the natural boundary, delimited by a dense set of singularities.). This would suggest that the STR and the region outside are two totally different domains, and there may not be a "true" tetrational that solves them both, or, if there is, it would not agree with the regular iteration. Which makes me wonder. There's this graph:
http://math.eretrandre.org/tetrationforu...926#pid926
which shows the possible region of convergence of Robbins' tetration, overlapping the STR a little. Could one test there and calculate it out really really really far and try to see if it does indeed disagree with the value of the regular iteration as would be expected if this hypothesis was correct?
Then again, the hypothesis could also be incorrect. I'm a little dubious about point #4: how does \( \exp^{t}_b(w) \) being non-analytic at w = a imply it non-analytic in *b* when w = *1*? Can you do a graph of the derivative \( \frac{\partial}{\partial b} \exp^{t}_b(1) \) for the regular iteration, in the same way that you did the graph of the regular iteration you posted here earlier?
http://math.eretrandre.org/tetrationforu...926#pid926
which shows the possible region of convergence of Robbins' tetration, overlapping the STR a little. Could one test there and calculate it out really really really far and try to see if it does indeed disagree with the value of the regular iteration as would be expected if this hypothesis was correct?
Then again, the hypothesis could also be incorrect. I'm a little dubious about point #4: how does \( \exp^{t}_b(w) \) being non-analytic at w = a imply it non-analytic in *b* when w = *1*? Can you do a graph of the derivative \( \frac{\partial}{\partial b} \exp^{t}_b(1) \) for the regular iteration, in the same way that you did the graph of the regular iteration you posted here earlier?
