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