imho a core issue

[sheldonison]

So Sheldon , did you find time to compute the results ?
Guess im a bit impatient , but i expected them sooner since you mentioned vacation time.
....

I've had some writer's block trying to decide how to proceed. Also, I have cleaned up some of the equations that I will eventually post, which will also include ways to calculate the Taylor series coefficients, and ways to calculate accurate approximations as n grows arbitrarily large (taylor series coefficients larger than 20 million). I have the numerical work done (for quite some time actually), but most of the numerical work I've done so far is on the basechange. I have the mathematical equations for tommysexp as well. The mathematical equations for Taylor series coefficients of TommySexp, tracks the taylor series of the basechange function, at twice the frequency, so that \( a_{2n}\approx{b_n} \) where \( a_n \) is for tommysexp and \( b_n \) is for basechange, for any value of \( n>\approx 7 \), both scaled by the appropriate singularity distance.

Also, there's two different ways to present the results, both of which turn out to be equivalently difficult and equivalently manageable. One is to generate the basechange function or the tommysexp function by iterating logarithms, and look how the taylor series change as you increase the number of iterated logarithms. But what is more interesting to me is to calculate the slog of superfunction of exp(z-1), which is equivalent to cheta(z)/e-1, or the slog of the superfunction of 2sinh(z). The equation which quickly becomes 1-cyclic as z increases is \( \theta(z)=\text{slog}(\text{superfunction}(z))-z \), where slog(z) is the inverse of the Kneser's sexp(z) function, which is known to be analytic. As z increases, theta(z) quickly converges and with theta(z+1)=~theta(z). theta(z)'s high frequency Taylor series coefficients change in the same way as tommysexp(z) or the basechange(z), and it is also nowhere analytic but infinitely differentiable. So I had a bit of writer's bloc in trying to decide which way to post the results, but I think the I'll probably do both, which will make the results that much more verbose. I need to decide what graphs to present, and then post all of it together, with the pari-gp program, and some graphs, and the equations, and methods to do the approximations. So that's the reason for the delay. Thanks for the interest. I'm in the process of adding numerical results for tommysexp for large values of n, though I have previously posted the taylor series coefficients for tommysexp(z) at z=0, for values of n up to several dozen. I will extend that to several hundred, showing the patterns, and then at the crossover where the next singularity takes over, which I've predicted to be around z^13million or z^14million. update got the program working for tommysexp, the frequency where the approximate radius of conversion switches from 0.45729 to 0.034659 is \( z^{1352619} \). The log of the taylor series coefficient is 1058342.36381, and the coefficient is negative. I was probably remembering wrong, off by a factor of 10, when I thought it was 13-14million. So now all I need to do is post the equations...
- Sheldon