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

[bo198214]

Perhaps f will even converge as n increases to infinity, just as the function converges on the real number line....

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.