Papers › The Limits of a Family; of Asymptotic Solutions to The Tetration Equation
The Limits of a Family; of Asymptotic Solutions to The Tetration Equation
James David Nixon
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In this paper we construct a family of holomorphic functions β_λ (s) which are solutions to the asymptotic tetration equation. Each β_λ satisfies the functional relationship β_λ(s+1) = (e^(β_λ(s)))/(e^(-λs) + 1); which asymptotically converges as logβ_λ(s+1) = β_λ (s) + 𝒪(e^(-λs)) as (λs) →∞. This family of asymptotic solutions is used to construct a holomorphic function tetᵦ(s) : ℂ/(-∞,-2] →ℂ such that tetᵦ(s+1) = e^(tetᵦ(s)) and tetᵦ : (-2,∞) →ℝ bijectively.
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