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.
Categories: math.CV math.DS