complex iteration

[Gottfried]

Jay,
I'm just rereading several posts of earlier threads, and came across this one.

The point is, fractional iteration for reals should be well-defined, so long as we limit the domain and range to reals. We should be able to prove the function exists and can be approximated with arbitrary precision. We might not, however, be able to find a power series which converges if we consider all terms in the series. But this doesn't prevent us from solving tetration regardless.

To illustrate the problem for the exp(x)-1 iteration I reactivated some older formulae and computed a table of coefficients which serve for a bivariate powerseries in x and h (the top- and iteration-parameter in the exp(x)-1 iteration). I think, that occuring divergences are not principally unsuited for numerical evaluation, as Euler-summation of some divergent series shows.
I just tried to reverse the summation in the bivariate powerseries using naive Euler-summation of terms, and get a reasonable result and also more insight in the character of divergence...
See
Coefficients for U-tetration

Gottfried