• Corpus ID: 233024768

The Limits of a Family; of Asymptotic Solutions to The Tetration Equation

@inproceedings{Nixon2021TheLO,
  title={The Limits of a Family; of Asymptotic Solutions to The Tetration Equation},
  author={James David Nixon},
  year={2021}
}
  • J. Nixon
  • Published 1 April 2021
  • Mathematics
In this paper we construct a family of holomorphic functions βλ(s) which are solutions to the asymptotic tetration equation. Each βλ satisfies the functional relationship βλ(s+ 1) = eλ e−λs + 1 ; which asymptotically converges as log βλ(s+1) = βλ(s)+O(e) as <(λs)→∞. This family of asymptotic solutions is used to construct a holomorphic function tetβ(s) : C/(−∞,−2]→ C such that tetβ(s+ 1) = eβ and tetβ : (−2,∞)→ R bijectively. 
Infinite Compositions and Complex Dynamics; Generalizing Schr\"{o}der and Abel Functions
Using infinite compositions, we solve the general equations $P(\lambda w) = p(w)f(P(w))$ for holomorphic functions $p$ and $f$. We describe the situations in which this equation is palpable; and

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follows by the induction hypothesis because φ (n+1)m (s, z) ⊂ N -it's in a neighborhood of A which is in N