Multiple exp^[1/2](z) by same sexp ?

[sheldonison]

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exp^[1/2](u) IS NOT DEFINED UNIQUELY BY THE SAME BRANCH OF SEXP THAT INCLUDES A
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I wasn't able to follow your logic -- too many variable names that seem to depend on each other. You have two previous threads on exp^{1/2}, which discussed the branch singularity at L, and the branch singularity at sexp(-2.5)=~-0.36+Pi i. Can you compare this result with the previous result, that showed exp^{1/2} has branches?

http://math.eretrandre.org/tetrationforu...hp?tid=849
http://math.eretrandre.org/tetrationforu...hp?tid=544

I do find it interesting that it seems there are no known entire functions with fractional exponential growth. I would define the exponential growth by the following equation, converging to a number bigger than zero, and less than one, for any arbitrary function f. For example, f=exp^{0.5} would have a growth rate of 0.5. The gamma function has a growth rate of 1, as does exponentiation to any base. The double exponential has a growth rate of 2. iterating x^2 or any finite polynomial has a growth rate of zero. Iterating a super-exponential function (there are many entire examples) would have an infinite growth rate.

\( \text{growth}_f = \lim_{n \to\infty}\frac{\text{slog}(f^{o n})}{n} \)

Perhaps this entire function growth rate question should be a conjecture? It seems it would be fairly easy to falsify, if one could find a counter-example.