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

[Ember Edison]

I have been reading post in forum for two weeks. Now I feel I was too young, too simple, and naive for tetration.
I need somebody to help me clarifiy the elementary knowledge for tetration.

My understanding of the tetration is:

Code definition:
\( \infty^* \)=ComplexInfinity (infinite magnitude, undetermined complex phase)
Not consider the branch cut:
\( tet_b(slog_b(z))=z \)
\( slog_b(tet_b(z))=z \)
\( tet_{sroot_h(z)}(h)=z \)

and:
\( tet_{conj(b)}(conj(z))=tet_b(z) \)
\( slog_{conj(b)}(conj(z))=slog_b(z) \)
\( sroot_h(conj(z))=sroot_h(z) \)

and:
\( tet_0(0)=1, tet_0(1)=0, tet_0(\infty^*) \)is oscillates  infinitely, but maybe 0 and 1 are different branch of the infinite iterated exponential.
\( tet_1(0)=1, tet_1(1)=1, tet_1(\infty^*)=1 \)
in other bases:
\( tet_b(\infty^*)=\frac{\mathrm{W_{cut}}(-\ln{z})}{-\ln{z}},cut\in\mathbb{Z} \)
\( sroot_{\infty^*}(z)=({\frac{1}{z}})^{-\frac{1}{z}} \)


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:
\( base=\pm\infty \), Andrew Robbins
base=0, not supported
\( base\in(0,e^{-e}) \), unknown
\( base\in[e^{-e},1) \), Koenig, no code
base=1, Andrew Robbins
\( base\in(1,e^{e^{-1}}) \), Koenig, fatou.gp
\( base=e^{e^{-1}} \), Ecalle, fatou.gp
\( Arg(base)\in({\frac{14\pi}{30}},{\frac{21\pi}{30}})\wedge({\frac{42\pi}{30}},{\frac{47\pi}{30}}),\left| base \right|<1.76. \)ill-region for Fatou.
other, Fatou, fatou.gp

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.

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.