How do I cite this document and does it say what I think it says?

[sheldonison]

So then what is the significance of Trappmann's formulas then if they don't let you write what the half exponential function actually does to a number? What that entire paper simply to suggest that there is a relationship between fractional heights and fractional iterations which is already a common-sense conjecture anyway?

Trapmann proves a uniqueness criterion for the inverse of tetration.  Given a tetration solution for a real base>exp(1/e), one can generate an alternative analytic solution which also meets the criterion TetAlt(z+1)=exp_b(TetAlt(z))
\( \text{TetAlt}(z)=\text{Tet}(z+\theta(z)) \)
where theta(z) is a 1-cyclic fourier series.  Trapmann proves than any such alternate solution will not meet his uniqueness criteria for the inverse of tetration, and will not behave as nicely in the complex plane as Kneser's solution.  Trapmann's proof works for the Abel function, or slog, or inverse of tetration.