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

[tommy1729]

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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 not feel those threads are strongly related.

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^{oz} 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.

Lets call that "conjecture entire 1".
Your intuition is correct.
And also it is strongly related to another recent conjecture of you ( or a repost of it ) :

\( \lim_{z \to \infty} \text{slog}(f^z)=\text{slog}(f^{z+1})-1 \)

(made in the slog_b(sexp_b(z)) thread)

Lets call that " conjecture entire 2"

Im very optimistic about both conjectures and dare to say a proof is 99% complete.
I have believed them for over 25 years.

As for " conjecture entire 1" I feel forced to notice
http://en.wikipedia.org/wiki/Weierstrass_product

Yes the famous weierstrass factorization theorem.

Together with induction that should be a strong tool in a proof.

Notice that the product expansions given in the wiki as an example ( sin and cos ) have MULTIPLE product expansions !

This might complicate the proof.

if every entire f(z) = z^a exp(g(z)) (1-z/a_n)
(product over positive integer n , a_n integers not necc all distinct , g(z) another entire function )

That would be simpler. But its not exactly like that ...
Maybe there is already a proof of it in the literature ?

" conjecture entire 2" has also been considered for proof by me. The issue here is the nondifferentiable nature of certain aspects.

one more thing about " conjecture entire 1"

In a recent thread I started , I asked for an entire function that grows like tetration. Lets say F(z)
By analogue you ask for an entire function that grows like exp^[1/2].
How about F(F^[-1](z)+1/2) as a solution to your problem ? Clearly F^[-1] cannot be entire ( it must have branches , being the inverse of a nontrivial entire function ).
Hence F(F^[-1](z)+1/2) is NOT the solution we want.

Maybe that helps.

regards

tommy1729