Calculating the fractional iterates of \(f(z) = e^z-1\) to reasonable accuracy

[Gottfried]

HEY! GOTTFRIED!

 
Well I'm chewing on your comment; but even more I'm still chewing to further understand
the inherent problems of that theta-construct and its evaluation.

The occuring of the increase of the coefficients which forces to evaluate the series only
as asmptotic is not really new for me, but it deconstructed ;-) my enthusiastic approach to
simply increase the number of coefficients for the polynomials in 'z' to 128...
Another problem with this is then the cpu-runtime consumption of many rational
coefficients with many many decimal digits ... if you are in the Pari/GP-user mailing
list you see one of my problems: how to convert my polynomials with rational
coefficients to such with real coefficients and limited precision to improve
runtime-characteristic (and first: to come into the near of your runtime speed). 

All this is primarily still for improving the intuition of your construction in detail & effect.

So at the moment: nothing new from my side, solving technical problems and
improving my test-environment to make visible what happens really.
I'll see whether I can do some reproduction of your plot, that shall be a cornerstone...

Kind regards -

Gottfried