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regular iteration of sqrt(2)^x (was: eta as branchpoint of tetrational) - JmsNxn - 06/09/2011
@ 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:
I understand \( \text{Lsexp}_{\sqrt{2}}(\text{Lsexp}_{\sqrt{2}}^{-1}(z) + \sigma) = \exp_{\sqrt{2}}^{\circ \sigma}(z)\,\,;\,\,\R(z) < 2 \)
and
\( \text{Usexp}_{\sqrt{2}}(\text{Usexp}_{\sqrt{2}}^{-1}(z) + \sigma) = \exp_{\sqrt{2}}^{\circ \sigma}(z)\,\,;\,\,\R(z) > 4 \)
Therefore how do we generate \( \exp_{\sqrt{2}}^{\circ \sigma}(z)\,\,;\,\,\R(z) \in (2, 4) \)?
Do we create a middle super function?
Secondly, why doesn't \( \text{Usexp}_{\sqrt{2}}(0) = 8 \)? Is there a reason it isn't like this? because if it was centered at 8 it would give the beautiful result:
\( \text{Usexp}_{\sqrt{2}}(\sigma) = 4\,\,\bigtriangleup_{\sigma}^{\small{\sqrt{2}}}\,\,4\,\,:\,\,\R(\sigma) \le 2 \)
It would also be consistent with the cheta function, where it's centered at 2 times the fix point.
RE: regular iteration of sqrt(2)^x (was: eta as branchpoint of tetrational) - bo198214 - 06/09/2011
(06/09/2011, 06:20 PM)JmsNxn Wrote: 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:
Here is some introduction into the topic.
RE: regular iteration of sqrt(2)^x (was: eta as branchpoint of tetrational) - sheldonison - 06/09/2011
(06/09/2011, 06:20 PM)JmsNxn Wrote: @ Sheldon's root 2 postings:Hey James,
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?
\( \exp_{\sqrt{2}}^{\circ \sigma}(z)=\text{Usexp_{\sqrt{2}}(\text{Uslog_{\sqrt{2}}(z)+\sigma) \)
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/tetrationforum/showthread.php?mode=linear&tid=260&pid=3296#pid3296
And another really good thread.
http://math.eretrandre.org/tetrationforum/showthread.php?mode=linear&tid=69&pid=534#pid534
RE: regular iteration of sqrt(2)^x (was: eta as branchpoint of tetrational) - JmsNxn - 06/09/2011
Thanks sheldon that was really helpful. Is there any code yet generating these two functions?
And my second question still stands, though;
why exactly does \( \text{Usexp}_{\sqrt{2}}(0) = 5.767053... \), is this point arbitrary? If it was shifted to 8 it wouldn't affect its status as a super function of root 2 would it? It's just a horizontal shift right?
RE: regular iteration of sqrt(2)^x (was: eta as branchpoint of tetrational) - sheldonison - 06/09/2011
(06/09/2011, 11:04 PM)JmsNxn Wrote: Thanks sheldon that was really helpful. Is there any code yet generating these two functions?
And my second question still stands, though;
why exactly does \( \text{Usexp}_{\sqrt{2}}(0) = 5.767053... \), is this point arbitrary? If it was shifted to 8 it wouldn't affect its status as a super function of root 2 would it? It's just a horizontal shift right?
Horizontal shift -- yes and no. It is what comes out of the limit equation, that generates Usexp(0), limit as n->infinity. I'll need to dig that equation out. But even if you do a horizontal shift, it gets cancelled out since Uslog is the inverse of Usexp. I do have the lower level primitives, "superf(z)" and a "isuperf(z)" functions in kneser.gp. Type init(sqrt(2)), and those two functions are available. For bases<eta, which also have a lower superfunction, I have also implemented superf2(z), and isuperf2(z).
- Sheldon
RE: regular iteration of sqrt(2)^x (was: eta as branchpoint of tetrational) - Gottfried - 06/15/2011
(06/09/2011, 06:20 PM)JmsNxn Wrote: Therefore how do we generate \( \exp_{\sqrt{2}}^{\circ \sigma}(z)\,\,;\,\,\R(z) \in (2, 4) \)?
Do we create a middle super function?
Beginning at x=1 we use the lower fixpoint (at 2) and the iterates are always between -infty and 2. if we begin at x=3 the iterates are always between 2 and 4. However, we can connect the two areas. If we begin at x=1 and iterate with the complex height h=0 + 2*Pi*i/log(log(2)) then we get exactly one value in the 2..4 interval. That is that we just switch the sign of the value of the schröder-function from positive to negative (one half round in the complex plane). That makes it also possible to define a "norm"-height for that values in the 2..4-interval. We set the real part of the height = 0 where x=1 was mapped to.
Gottfried