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Can we get the holomorphic super-root and super-logarithm function?
(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 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.
- Sheldon

Messages In This Thread
RE: Can we get the holomorphic super-root and super-logarithm function? - by sheldonison - 06/09/2019, 04:31 PM

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