complex iteration

[Gottfried]

I'm starting to think the problem is complex iteration. If you take the iterating function as a function in two variables, n and z, then its radius of convergence is limited by both variables. We can fix z, and explore the radius of convergence for n.
I'll need to investigate this more, but non-complex iteration counts would seem the be the source of the otherwise unseen singularities.

I'll coin approximations for some complex and irrational values:

Code:

:
    f: f(x)= exp(x)-1
    z = s'th iterate of f(1)

Code:

Approximations by Euler-summation
   s= I      z ~ 0.847025032777 + 0.373003870989*I
   s=-1      z ~ 0.693147180560 + 0             *I   (log(2))
   s=-I      z ~ 0.847025032777 - 0.373003870989*I
   s= 1      z ~ 1.71828182846  + 0             *I   (exp(1)-1)
   s=1/2+I   z ~ 0.996846811341 + 0.546746701133*I
   s=sqrt(2) z ~ 2.37133411

64 coefficients, approximation by Euler-summation of
different orders (also separately for real and imaginary part)


Gottfried