The upper superexponential

[sheldonison]

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Unfortunately the convergence gets quite bad for \( b \) approaching \( e^{1/e} \), so I could not really check numerically.
On the other hand Walker describes also two solutions for \( b=e^{1/e} \) in "On the solutions of an Abelian equation". I did not really read this article, but I think he also showed that these solutions are not the limit of approaching \( e^{1/e} \).
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in the thread regular slog.
Literature is: Szekeres "Regular iteration of real and complex functions."

Kouznetsov has graphs of the lower super exponential for \( b=e^{1/e} \) in the citizendium wiki. He says "the function approaches its limiting value e, almost everywhere". I haven't seen any graphs for the upper superexponential though.

For \( b>e^{1/e} \), the function exponentially decays to its limiting value in the complex plane at +/- i \( \infty \). This is probably also true for the upper super exponential for \( b=e^{1/e} \), as the value at the real axis increases ...