imho a core issue

[sheldonison]

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4 : it ( basechange ) is not proven to not be analytic

5 : even if basechange is not analytic , how about 1 : radius of convergeance ?

Agree, its not proven. It is proven to be infinitely differentiable, and I want to show that the taylor series coeficients grow fast enough that the function has a zero radius of convergence.

6 : if two functions are similar in property or close in value that does not imply they must both be analytic or non-analytic.

I agree, and intend to treate sexp2sinh(z), which is what I had been calling tommysexp(z), completely independently, but using the same methods.

7 : as i often say , i compute it slightly different which might effect the millionth digit or the analytic properties , you need to understand that if you use a variant your properties are for that variant also.

8 : there are not enough singularities close enough to the real line to make my function exp^[1/2](x) NOT analytic in x near the real line x.
( i use analytic continuation to go to the complex plane )

9 : when you talk about non-analytic do you perhaps mean the other parameter z instead of x in exp^[z](x) ??
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I'm working on a draft post with the equations. As to remark (9), my focus is would be on iterating logarithms of the superfunction of 2sinh(z),
define g(z) as the superfunction of 2sinh(z)
\( g(z+1)=\exp(g(z))-\exp(-g(z)) \)
then,
\( \text{sexp2sinh}(z)=\lim_{n\to\infty}\log^{[n]}g(z+n+k) \)
My understanding is that sexp2sinh(z)=tommysexp(z), but maybe not, so sexp2sinh(z) is a better name to use for this alternative sexp(z) function. The sexp2sinh(z) function, as defined above, was inspired by our exchanges on the tetration forum concerning tommysexp(z).

My interest is in exploring the derivatives of sexp2sinh(z), and the basechange(z) function, which are both well defined at the real axis, and both look like tetration. Both functions have very interesting taylor series representations, whose coefficients appear to eventually grow faster than any analytic function, but do so in a most unusual way. The draft I am working on will show how I approximate the taylor series coefficients for these functions, in the hopes that this might be a step towards a rigorous proof that these functions are nowhere analytic.
- Sheldon