03/22/2010, 10:14 AM
Hmm. This suggests there are two quite distinct approaches to the tetration using fixpoints, each of which covers one of two seemingly vastly different domains. Namely, we have the Shell-Thron region wherein the regular iteration is used, which yields a solution that is real valued at the real axis, but this solution has (may have? Still need more rigorous proof) a natural boundary at the region border, so it cannot (might not?) be extensible outside said region. Outside that region, we have the rest of the plane, for which the extension would be achieved via the bipolar method, which may not be extensible inside the STR, or if it is, it cannot be real-valued for \( 1 < b < e^{1/e} \).
(BTW, I've been playing around with another tetration method based on trying to use the Borel summation on Ansus' continuum-sum formula. If you want, I can post some rough observations from an attempt at numerical approximation. I'm still not sure if it converges, as it seems to take tons of precision and terms to work, so I can't really press past more than a few decimals of accuracy.)
(BTW, I've been playing around with another tetration method based on trying to use the Borel summation on Ansus' continuum-sum formula. If you want, I can post some rough observations from an attempt at numerical approximation. I'm still not sure if it converges, as it seems to take tons of precision and terms to work, so I can't really press past more than a few decimals of accuracy.)
