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+--- Thread: Can we get the holomorphic super-root and super-logarithm function? (/showthread.php?tid=1211)
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RE: Can we get the holomorphic super-root and super-logarithm function? - Ember Edison - 06/10/2019
(06/09/2019, 04:31 PM)sheldonison Wrote:I know 1 and 0 is singularity base, i just want to look the function behavior when the bases very close 1 or 0...(06/09/2019, 04:03 PM)Ember Edison Wrote: Thank you for your work.It's very helpful.
by the way...I can't set the base with -0.1 <= x <= 0.62, 0.88 <= x <= 1.042, how can i do something to fix it?
Instead of b, try b+1E-25*I so that its not on the real axis; that might help a little. But if using fatou.gp for bases close to zero, the sexpinit breaks since the real part of the pseudo period of the fixed points gets too small so the theta mapping falls apart. base=0 is a singularity. For real bases>1 and less than 1.042, the problems are different as the imaginary period gets smaller and smaller. base=1 is also a singularity. Its probably best to accept these as the unavoidable computation limits. So then the superroot problem mostly has the same computation limits as extending Kneser to complex bases and bases<eta.
edit: Its been awhile since I've looked closely at the singularity for circling around the base=1. sexp(-0.5) is complex valued for bases<1, but the branch is more significant than for 1<b<eta where both fixed points are still real valued. It is because the secondary fixed point is no longer real valued. for sexpinit(0.7) the two fixed points are L1~=0.762 and L2~=-7.3505+11.632i. Kneser's solution requires us to weave together the two Schroder/Abel functions from the two fixed points. This is done with a 1-periodic theta mapping, z+theta(z). This this allows the complex conjugate pair of fixed points for real bases>eta to generate a real valued tetration function. Perhaps I will post more later.
I thing we need use set theory to define the base=0 and base=1, and not complex analysis.
The article[1] say if we use very small w get tet(b+w*I,z) to close tet(b,z) , "will be very slightly different for real" base when 1 <= b <= etaB, so I'm very very worry about the other real base will happen again. (example of course is 0 <= b <1).
The article[1] say he can work "except if we are too close to either b = 0, b = 1, or b =" etaB. I think we can close the bases 1 or 0 like we close the etaB if the article is true.
Ps:You use more exp() to evaluate more Re(height) >= 1. So can we use more log() to evaluate more Re(height) <= -1?
Ps2:Why we need to avoid etaB? When low precision, we can use sexpeta/slogeta to process singularity at etaB in my imagination.
We should to avoid the singularity at 0 and 1, if the sexpeta/slogeta can use.