using sinh(x) ?

[bo198214]

Ok, first let us verify that it is indeed an iteration of exp, i.e that indeed satisfies:
and .

neglecting some rules of properly evaluating limits we get
.

and
because towards infinity gets arbitrarily close to .

Basically thats the iteration equivalent of the Abel function Lévy proposes:

where is the Abel function of (or in Lévy's case ).

The superfunction is then (the inverse of ):



which is the same as Tommy's superfunction.

We can do something similar with not only or but with any function that does not deviate too much from exp at infinity (i.e. all functions such that ).

as for the ROC i assume a plot for increasing n says more than a thousand words.

plot n = 1 -> 100. z = 1/2

I think you were not attentive when proposing to compute , already can not be computed in even sage's multiple precision arithmetic.