How do I cite this document and does it say what I think it says?

[sheldonison]

... We want to know what raising something to the "1/2" height means, at least according to this conjecture, and z in this case is the height of tetration:

\( \exp_b^{ \frac{1}{2}}(1)=s \exp_{b}(\frac{1}{2})= \underbrace{b^{b^{b^{...}}}}_{\text{\frac{1}{2} times}} \)

How should this result be interpreted to yield a final answer in terms of standard mathematical functions for b=e?

This is a very good question.  William Paulsen's paper from earlier this year, "figure 1" would be a good starting place;


You might also want to view my post showing pictures of the Riemann mapping: https://math.eretrandre.org/tetrationfor...p?tid=1172 But this figure is hard to explain; the short explanation of this figure is as follows:

• Start with the Schroeder function for the complex fixed point L~=0.318131505 + 1.33723570I; then e^L=L
  
• Turn the Schroder function into an Abel function, Abel(exp(z))=Abel(z)+1.  
  
• But this is a complex valued Abel function for the real number line!
  
• It has singularities at z=0,1,e,e^e....
  
• Take the Abel function of the real axis ... then you get the repeating function above in the darkened line, along with the requisite singularities, starting with -infinity
  
• Now wrap the repeating 1-cyclic pattern around the real axis via the mapping \( y \mapsto \exp(2\pi\i z) \)
  
• Take the Rieman mapping of that circle function so that it maps to the unit circle.
  
• Put the singularity at z=1.  Now, z=-1 corresponds to 0.5, exp(0.5), exp(exp(0.5)) etc.  Unwrap it onto the original drawing, and magically you have a real valued tetration of the real axis.
  

Unfortunately, the Riemann mapping is hard to compute, so realistically, all that we have is the proof of the existence of a nicely behaved solution.  In general, Riemann mappings cannot be expressed in terms of elementary functions or series and therefore Kneser's solution probably isn't expressable in terms of elementary functions. 

As far as I know, nobody has a rigorously proven computation technique published; Paulsen's paper gives numerical results just like Trapmann's paper, along with heuristic arguments for why it should equal Kneser's solution.  There are a few really good programs that can easily calculate sexp(-0.5) for any base, like my pari-gp program, fatou.gp   In the future, I expect someone will show how to make one of these computation schemes mathematically rigorous ...