Complex Iterations of Entire Functions

In a previous paper we produced a complex iteration of a holomorphic function $\varphi$ 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 $\varphi$ 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 ${\varphi }^{\circ z}(\xi )$ for $\xi$ in a the immediate basin of a geometrically attracting fixed point involves an expression using the natural iterates ${\varphi }^{\circ n}(\xi )$, 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 $0<\alpha \le e^{1\mathrm{/}e}$ to arrive at the multi-valued function ${(}^z\alpha )$.