The Bounded Analytic Hyper-operators

Abstract

In a previous paper \cite{ref1} we produced a sequence of analytic functions $\{\alpha {\uparrow }^{n}z{\}}_{n=0}^{\infty }$ when $1\le \alpha \le e^{1/e}$ and $z$ was in the right half of the complex plane, the \emph{bounded analytic hyper-operators}. This was a sequence of functions where each function was the \emph{complex iteration} centered about $1$ of the previous function. We show $\alpha {\uparrow }^{n}z$ is holomorphic and bounded for $\Re (z)>1-n$. We give a closed form expression for an analytic function in all variables $\alpha {\uparrow }^{s}z$, with $1<\alpha <e^{1/e}$, $\Re (s)>1$ and $\Re (z)>0$. This three variable function, when restricted to the real line; $\alpha {\uparrow }^{t}x$ when $x,t\in {\mathbb{R}}^{+}$ and $t\ge 1$; has initial conditions $\alpha {\uparrow }^{t}1=\alpha$ with $\alpha {\uparrow }^{1}x={\alpha }^x$ and satisfies the functional equation $\alpha {\uparrow }^t(\alpha {\uparrow }^{t+1}x)=\alpha {\uparrow }^{t+1}(x+1)$.

arXiv commentThe method of proof is improperly worded, the result is being reproved in a slightly more aesthetic manner, and being more clearly worded

References in corpus (1)

• Complex Iterations and Bounded Analytic Hyper-operators