Asymptotic Solutions of the Tetration Equation [article]

James David Nixon
2022 arXiv   pre-print
In this report we construct a family of holomorphic functions β_λ,μ (s) which behave asymptotically like iterated exponentials as |s| →∞ in the right half plane. Each β_λ,μ satisfies a convenient functional relationship with nested exponentials; and has a series expansion that converges in a half-plane. They provide a nearness to the dynamics of the map e^μ z : ℂ→ℂ and behave asymptotically as a fractional iteration would behave. These objects are used to describe the various orbits of the
more » ... ential function. We describe where Abel equations are feasibly constructed from β. Where there exists wildly holomorphic functions with period 2 π i / λ that are holomorphic Abel functions of the form t(s+1) = e^μ t(s).
arXiv:2208.05328v1 fatcat:5gb4smhekbddjgbozbycz7gz5m