2015/11/30 by James David Nixon, James D. Nixon, Nixon, James D.
Mathematics · #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Equations Stability Results #Mathematical Dynamics and Fractals #Mathematical and Theoretical Analysis #math.CV #math.DS
paper · pdf · doi:10.48550/arxiv.1512.00754
A crucial error was found in the domain of functions in which the result applies, it is being altered to accomodate this new error
openalex publication_date 2015/11/30 · arxiv created 2016/04/06 · arxiv updated 2016/04/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In a previous paper we produced a complex iteration of a holomorphic function ϕ in the immediate basin of a fixed point whose multiplier is a real number and in between zero and one. We further explore this problem, allowing the multiplier to be a complex number and its modulus to be in between zero and one. We find expansions of results derived before. We will continue to use Ramanujan's master theorem and the process of factoring appropriately bounded exponential functions by their values on the natural numbers. We will obtain all complex iterations of entire ϕ in the immediate basin of a fixed point whose multiplier is inside the punctured unit disk. The evaluation of any branch of the complex iteration ϕ∘ z(ξ) for ξ in a the immediate basin of a geometrically attracting fixed point involves an expression using the natural iterates ϕ∘ n(ξ), the fixed point ξ0 and its multiplier. We will give applications on certain bases of tetration functions. Namely we will iterate the principal branch of αξ for 0 < α≤ e1/e to arrive at the multi-valued function (z α).