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

[sheldonison]

Ps: How to get the photo like you in pari/gp? I need some plot code. 
https://math.eretrandre.org/tetrationfor...p?tid=1017
ok, i see MakeGraph().But why i use write() nothing in my test file?
Ps2:How far are we get a perfect code to evaluate super-roots if we get the perfect tetration code?

Thanks for the links.  I had seen Paulson's javascript code before, when I exchanged emails with the author last May.
I'm assuming you got MakeGraph to generate plots;   This MakeGraph example takes 20minutes???
sexpinit(exp(1));
fmode=3; /* f(z) is used by MakeGraph; fmode=3 sets f(z) to safesexp(z) */
MakeGraph(1080,540,-3,3,9,-3,"sexp_e.ppm",1); /* from -3 to +9 at real axis, -3 to +3 imaginary */
write ("foo.txt","hello");
pari-gp is really slow writing one pixel at a time; attached is a faster version of MakeGraph; I also have faster sexp inversion code, but the faster sexp code has bugs with some complex bases, so I didn't include it.  
  MakeGraphSpeedup.gp (Size: 3 KB / Downloads: 998)  

I think one needs to start by understanding the inverse of super-root before one can understand super-root since you need to take the inverse of sexp_b(n) for some particular value of n for which the super-root is interested in, so I would think you want to understand how sexp_b(1+0.5*I) behaves for example, for the variable b.  What if n is between 0 and 1?  Is sexp_b(0.5) bounded as base(b) changes?  I have no idea.  So obviously, I don't understand the inverse of the function, much less have a perfect code ...  My suggestion is to start with Newton's method and iterate on finding approximations for the inverse, but Newton's method relies on a guess being close enough to the correct answer to converge.  For some particular value of n, I think I might know how to numerically compute the Taylor series for the function given some starting point since I had done something like that earlier to show complex tetration is analytic in the base.