07/13/2019, 06:36 PM

(07/11/2019, 04:57 PM)sheldonison Wrote:(07/01/2019, 12:11 PM)Ember Edison Wrote: I have been reading post in forum for two weeks. Now I feel I was too young, too simple, and naive for tetration.It looks better. I would like to post more, but time is limited. For sroot, all of the extensions need to be in the same analytic family.

I need somebody to help me clarifiy the elementary knowledge for tetration.

My understanding of the tetration is:

Quote:Code definition:

=ComplexInfinity (infinite magnitude, undetermined complex phase)

Not consider the branch cut:

and:

and:

is oscillates infinitely, but maybe 0 and 1 are different branch of the infinite iterated exponential.

in other bases:

tetration, super-root and super-logarithm is infinitely differentiable. (but I wasn't find code take the derivative...)

If the bases is hyperbolic, there is only one "regular" super-function. If the bases is parabolic, will have at least 2 "regular" super-function.(Leau-Fatou-flower)

The branch cut for super-function is infinitely.

fatou.gp will use all "regular" super-function to refactoring tetration.

bases regions for tetration:

, Andrew Robbins

base=0, not supported

, unknown

, Koenig, no code

base=1, Andrew Robbins

, Koenig, fatou.gp

, Ecalle, fatou.gp

ill-region for Fatou.

other, Fatou, fatou.gp

Kneser analytic solution can be extended to complex bases, and creates a family of complex base solutions, but

such solutions have singularities at bases like base=0,1, eta so one cannot talk about analytic base=0 or base=1 or base=exp(1/e) for Kneser. One cannot use the Koenig/Schroeder solutions or the Ecalle solutions in the Kneser construct for an sroot family.

So Schroeder contribute 1 sroot, Kneser contribute 1 sroot, 3 Singularity contribute 3 sroot and "infinity" contribute 3 sroot(Complex analysis, Ordinal arithmetic and generalized continuum hypothesis).

It's fucking cool.