regular iteration of sqrt(2)^x (was: eta as branchpoint of tetrational)

[sheldonison]

@ Sheldon's root 2 postings:

Hmm, this is all very interesting. I myself am interested in \( \text{sexp}_{\sqrt{2}}(z) \); since I think this may be the natural base semi-operators work in. Therefore I have a few questions:
....
Therefore how do we generate \( \exp_{\sqrt{2}}^{\circ \sigma}(z)\,\,;\,\,\R(z) \in (2, 4) \)?
Do we create a middle super function?

Hey James,
\( \exp_{\sqrt{2}}^{\circ \sigma}(z)=\text{Usexp_{\sqrt{2}}(\text{Uslog_{\sqrt{2}}(z)+\sigma) \)
The short answer, is that these functions are imaginary periodic. Usexp(z) has a period of approximately 19.236i. USexp is real valued at the real axis going from 4+delta to infinity. But at exactly half that period, the Usexp(z) function is also real valued from -infinity to infinity, gently making a transition from 4-delta to 2+delta. Lsexp(z) has a period of about 17.143i. And at exactly half that period, the Lsexp(z) function is real valued from -infinity to infinity, also gently making a transition from 4-delta to 2+delta. In going from 4-delta to 2+delta, these two functions can be lined up, so that they are nearly identical, but they differ by a tiny amount!

So, for \( z \in (2, 4)\,\; \Im(\text{Uslog_{\sqrt{2}}(z))\approx9.62i
\). Hope that helps.
- Sheldon
Here are some more links (to go on wiki page?)
http://math.eretrandre.org/tetrationforu...96#pid3296
And another really good thread.
http://math.eretrandre.org/tetrationforu...534#pid534