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 ${\displaystyle \beta_\lambda(s+1) = \frac{e^{\beta_\lambda(s)}}{e^{-\lambda s} + 1}}$; 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.