  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.
