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

[sheldonison]

... So,Can we get the holomorphic super-root and super-logarithm function and all parameter is complex?

https://math.eretrandre.org/tetrationfor...p?tid=1017
fatou.gp implements Kneser's super-logarithm or inverse of tetration for complex bases and complex heights.

There is a proof of uniqueness of Kneser's slog if the slog between the two primary fixed points is defined, unique and one to one mapping in a region bounded by a sickle between the fixed points.  One side of the sickle is a curve connecting the two fixed points, and the other side is the exponent of the curve.  The algorithm for fatou.gp is a bit complicated, but it compute's Kneser's slog for a very wide range of real and complex bases.

I have reviewed other papers by the author, but not this one.  The results from the other paper match fatou.gp exactly (within error limits), so I'm pretty sure that my program would also be the inverse slog function matching the author's results for complex base tetration as well.  There is only one valid extension of Kneser's solution to complex tetration bases.

He use Cross Track method to evaluate tetration in real bases, larger than e^(1/e), Your's method to evaluate bases e^(1/e), Sword-Track to evaluate Shell Thron boundary, Double Dagger Track to Evaluate other complex bases.

I am reading your super-logarithm code,it look like can evaluate all complex bases. Thank you for your work.
So now what conclusion can get with complex super-root?

I don't have access to a university right now, so I haven't downloaded Paulson's latest paper; I have downloaded two of his other papers.  I would guess that it would be nice to show formally that one of these algorithms rigorously converge to Kneser.  That's a complicated problem; whenever I think I'm close to being able to rigorously prove fatou.gp actually converges, I get sidetracked.  

Also, fatou.gp struggles with bases on the Shell Thron boundary since one of the fixed points is neutral and the Schroder function may not converge so there is no theta mapping so my algorithm converges poorly for bases on the Shell Thron boundary.  For bases on either side of the boundary, inside or outside the Shell region, my algorithm still works good, but it slows down quite a bit 
sexpinit(2+1.1*I) works good
sexpinit(2+1.2*I) works good
sexpinit(2+1.15*I) initialization slow; takes 40 seconds; normal precision 34 decimal digits
sexpinit(2+1.16*I) base too close to the Shell Thron boundary; no upper theta mapping; precision 14 decimal digits

I haven't worked with super-roots, and I don't have a super root algorithm for real or complex bases.  The function \( f(b) = \text{Tet}_b(0.5i)\;\; \) is analytic in the teration base b, and instead one may use other values like n=0.5i; or n=4; or n=2.5+0.25i or any other value of interest. Then \( f^{-1}(z) \) is the super-root for the value n in question and is also analytic.