Cyclic complex functions and uniqueness

[Kouznetsov]

In the case of complex domain, the uniqueness of tetration \( F \) seems to be provided by the axiom about the asymptotic behavior of \( F(z) \) at \( \Im(z) \rightarrow +\infty \):

So that would imply that any holomorphic function periodic on the real axis goes to infinity on the imaginary axis. Is that true? (My knowledge about complex analysis is not that exhaustive.)

Not so easy. The function can be doubly-periodic. However, the doubly-periodicity may be also good; if it breaks the postulate about the asymptotic behavior of tetration, then it can be used to prove the uniqueness of the solution.
You may expand the function into the Fourier series, and analyse the behavior at \( {\rm i}\infty \). For many functions, the series diverge, which indicates singularities. Those, that converge, may show exponential growth and be "scracthed" in the sense that at the plot, in the interval of length of unity, there are so many equilines, that the plotter fails to plot them.
Tetration \( F(z) \) is not entire function, because it has singularities and cut at \( z\le -2 \); in the paper mentioned, I write "indicates" instead of "gives the proof". In order to apply the expansion to the tetration, you may use the asymptotics \( F(z)=L + {\mathcal O}\Big( \exp( L z ) \Big) \). You may try to get the next term in the expansion; it seems, that \( F(z)=L + \exp( L z +Q) + {\mathcal O}\Big(\exp(2 L z ) \Big) \), where \( Q\approx 1.10214+1.54047{\rm i} \). The truncated asymptotic is entire function.
See also discussion at http://math.eretrandre.org/tetrationforu...37#pid1837.

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