Real and complex behaviour of the base change function (was: The "cheta" function)

[sheldonison]

Walker showed a similar convergence for \( f_n=\operatorname{dexp}^{[-n]}\circ \exp^{[n]} \), where dexp(x) = exp(x)-1.
He showed that the limit is infinitely differentiable on the real axis.
That means that he also wasnt clear about the complex behaviour otherwise he would have shown that the limit is holomorphic as a consequence of local uniform convergence.
But he could prove that local uniform (or compact) convergence only on the real axis, which does not suffice to imply holomorphy (because it could be that during the convergence non-real singularities get dense towads points on the real axis). I will persue this topic in the next days and have still some unexplored ideas at my hands.

Henryk, Do you have a reference for showing that the base change function is infinitely differentiable? Perhaps Walker's paper? I am now able to generate approximations for the Taylor series coefficients for the base change function. This would lead to a way of showing that the function is infinitely differentiable at the real axis, since all of the coefficients converge. But it would also lead to a way to show that the function is nowhere analytic, since the coefficients for large enough n eventually grow faster then any exponential. I wanted to read Walker's paper to see what his approach was, before posting my results.
- Sheldon