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

Abstract

In this paper we construct a family of holomorphic functions ${\beta }_{\lambda }(s)$ which are solutions to the asymptotic tetration equation. Each ${\beta }_{\lambda }$ satisfies the functional relationship ${\beta }_{\lambda }(s+1)=\frac{e^{{\beta }_{\lambda }(s)}}{e^{-\lambda s}+1}$; which asymptotically converges as $\log {\beta }_{\lambda }(s+1)={\beta }_{\lambda }(s)+\mathcal{O}(e^{-\lambda s})$ as $\Re (\lambda s)\to \infty$. This family of asymptotic solutions is used to construct a holomorphic function ${\text{tet}}_{\beta }(s):\mathbb{C}/(-\infty ,-2]\to \mathbb{C}$ such that ${\text{tet}}_{\beta }(s+1)=e^{{\text{tet}}_{\beta }(s)}$ and ${\text{tet}}_{\beta }:(-2,\infty )\to \mathbb{R}$ bijectively.

54 pages; 26 figures; I updated the coding section with more accurate numerical evidence. This a far more developed version than the second iteration of this paper; we've included graphs of the various tetration functions, and the actual tetration we want. We also tightened much of the bounds in our construction, and elucidated more on why we have convergence

References in corpus (3)

• The Compositional Integral: The Narrow And The Complex Looking-Glass
• Hyper-operations By Unconventional Means
• Second Order Transfer Equations; and Generalizations to Arbitrary Orders

Cited by in corpus (1)

• Infinite Compositions and Complex Dynamics; Generalizing Schröder and Abel Functions