Complex Iterations of Entire Functions

TL;DR

This paper extends the theory of complex iterations of entire functions within the immediate basin of fixed points, providing explicit formulas for all such iterations when the multiplier is inside the punctured unit disk, with applications to tetration.

Contribution

It generalizes previous results to complex multipliers, offering explicit expressions for complex iterations of entire functions in the immediate basin of fixed points.

Findings

Derived all complex iterations of entire functions with multipliers in the punctured unit disk.

Provided explicit formulas involving natural iterates and fixed point multipliers.

Applied results to principal branches of exponential functions to define multi-valued tetration.

Abstract

In a previous paper we produced a complex iteration of a holomorphic function $\phi$ 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 \emph{factoring} appropriately bounded exponential functions by their values on the natural numbers. We will obtain \emph{all} complex iterations of entire $\phi$ 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 $\phi^{\circ z}(\xi)$ for $\xi$ in a the immediate basin of a geometrically attracting fixed point involves an expression using the natural iterates $\phi^{\circ n}(\xi)$, the fixed point $\xi_0$ and its multiplier. We will give applications on certain bases of tetration functions. Namely we will iterate the principal branch of $\alpha^\xi$ for $0 < \alpha \le e^{1/e}$ to arrive at the multi-valued function $(^z \alpha)$.